"Any American woman in Saudi Arabia who wants to leave will be free to do so."

So spoke Prince Saud al-Faisal on Sunday afternoon, according to Indiana Republican Dan Burton. The setting was a hotel room in Jeddah, in a meeting with a U.S. congressional delegation there to press the cases of Americans who remain trapped in the kingdom there by Saudi law, which does not allow women or children to leave without the permission of a husband or father.

At a time when the Saudis are desperate to improve their image in the U.S., and spending millions to do so, their continuing failure to produce tangible results on something so obviously in their interests is not a happy sign that this is a regime willing to take on the Wahhabi view of the United States. What we've learned from the last week is that the House of Saud is willing to embarrass an American president even after he's complained that the Saudis are not handling these cases the way they should.

Prince Saud's words may sound encouraging, but in this desert kingdom what may look like progress all too often turns out to be a mirage. Give them their due: The Burton delegation has extracted more on this issue from Saudi Arabia in a few days than the State Department has managed in 15 years. But though American women may now, in theory, be allowed to leave (the prince said nothing about their children), not a single American is in fact getting out. And the fantastic developments and charges that swirled around this delegation's visit raise more questions than answers about the Saudis' good faith.

"That was my worry from the start," says Miriam Hernandez, an American women whose daughter, Dria Davis, escaped from Saudi Arabia years ago and testified before Mr. Burton's House Government Reform Committee back in June. "The delegation is on Saudi soil and on their terms, and the Saudis just want to engineer things to get the press off their backs."

Certainly the Saudis are not giving much on the ground. Even Prince Saud, says Mr. Burton, stated that Saudi Arabia did not recognize U.S. law on these issues, a statement endorsed by their actions over the years. In addition, on two of the highest-profile cases the congressmen were asking about--that of Alia and Aisha Gheshayan, daughters of Pat Roush, and of Amjad Radwan, Monica Stowers's daughter--huge snags materialized out of nowhere.

Take the Gheshayan sisters. Without warning, at the exact moment the congressional delegation was in Saudi Arabia seeking answers, the two women suddenly turn up in London, with a whole entourage of Saudi men including their husbands. There, they meet with U.S. consular officials and say they do not want to go to America. Dria Davis knows this game all too well. When Dria was 11, her father refused to let her return to her mother in America. Like the Gheshayan sisters, she too was brought before a U.S. consular officer, who wrote up a report saying Dria seemed cheerful and content with her life in Saudi Arabia. Quite different from the tapes of Dria screaming for help which were played before Mr. Burton's hearings. Dria was one of the fortunate ones: She escaped.

"Those girls [Alia and Aisha Gheshayan] were probably told what to say," she says. "My father told me he'd have me killed if I brought up my mother or said I wanted to leave. I think if these girls only saw their mother, they would cry and say they want to come home. They are not free."

Pat Roush has had 15 years of this. When her daughters were first taken to Saudi Arabia, the office of the governor of Riyadh participated in the making of a videotape--in the presence of a U.S. foreign service officer--in which the girls were brought out and said "I hate the United States" and "My mother hates me and my sister."

"What are the Saudis afraid of?" Ms. Roush says. "Why can't they come to America and tell it to a mother who's seen them only once, for two hours, in 15 years? If they really didn't want to stay in America, who could stop them?"

Amjad Radwan's case is equally murky. A week ago in Texas, President Bush specifically mentioned the case of this 19-year-old to Prince Bandar, the Saudi ambassador. But literally on the day Miss Radwan was to pick up a U.S. passport, she ran away from her mother's home, announced that she was married to a man her father had selected and through tears said that though she did still want to go to America, she was not ready to leave just yet.

Miss Radwan's mother describes her daughter as a teenager who was horribly abused as a child and has cracked under the emotional stress, a young woman whose father preyed on her insecurities and told her how she'd be hated and never make it in America. Miss Radwan did get a response: in theory, Miss Radwan now has a visa to leave whenever she wants, but of course no one knows what will happen if she ever tries to use it, gets pregnant by her new husband, etc. And it took a specific request by the president of the United States to get this far. "I know this is confusing to everyone," says Ms. Stowers. "My daughter got suckered into a relationship that will destroy her eventually and the only ones who have benefited are the Saudi PR guys who will continue about their merry way."

In short, what we have is a gigantic mess, where no one can speak for sure of the barest facts: whether Amjad Radwan really is married, how many children the Gheshayan sisters might have, etc.

The Saudis haven't helped. On Saturday, this writer received a phone call from the Saudi Embassy's Adel al-Jubeir, disclosing that the Gheshayan sisters were in London. Mr. al-Jubeir said that he himself hoped that the women would go to America and confront their mother and settle this once and for all--but that Saudi Arabia cannot tell its private citizens what to do. When I suggested that the Saudis managed to get them to London (at precisely the moment a congressional delegation is in Saudi Arabia), he said the sisters were "on vacation."

Ms. Stowers indicates that the Saudis are all too willing to twist arms if necessary. Two weeks before her daughter ran away, Ms. Stowers told me that her ex-husband had been taken to meet with "the princes," where he was threatened with losing his job if he didn't sign an exit visa for his daughter. Miss Radwan said he was very shaken up, but continued to tell her he'd "burn in hell" if he ever agreed to let her leave. There's not a single person on either the U.S. or Saudi side who doesn't believe that when the Saudis want to do something, they are perfectly willing to make their citizens offers they can't refuse.

Nor are these the only disquieting notes. When Mr. Burton was talking with Miss Radwan at a Riyadh Starbucks, they had their meeting interrupted by a member of the Saudi religious police. Uglier still, late that night, after that meeting, U.S. Ambassador Robert Jordan summoned the congressmen to tell them he'd heard from the Saudi foreign ministry that Miss Radwan and her husband had said Mr. Burton had offered her $1 million to get on their plane out of Saudi Arabia. Though Mr. Burton is careful to say that Prince Saud did not himself accuse Mr. Burton, the foreign minister did repeat the alleged bribe three or four times during his meeting with the delegation. The Saudis have their own hardball.

And then there are the other cases the delegation was looking into. Fairly typical is the one of an American woman who meets her Saudi husband at a U.S. college and gets married. When they divorce--again in the U.S., mind you--the wife seeks sole custody of the children; she pleads with the judge not to grant her "ex" unsupervised visits because he'll kidnap them to Saudi Arabia.

The Saudi gives the judge his word he will not do that, and the judge even takes the step of informing the Saudi Embassy in Washington that this man was not to be given passports to take the kids out of the country. Of course, as soon as he had his children on a two-week unsupervised visit they all turn up in Saudi Arabia, with the obvious connivance of the Saudi government. Since then, the mother has had no contact with her children.

The Saudis like to say it's not all one-sided. Mr. al-Jubeir, for example, was the second Saudi official to remind me that Americans kidnap their children too. That is true. But with a key difference. Children with their American parent in, say, Peoria are not calling their fathers in Riyadh screaming that they are in hell and want to get out before they're killed. American courts, moreover, have been most willing to give fathers visiting rights; in almost all cases the root of the problem is a Saudi father who, after being married and divorced under American law, violates these custody rulings.

Not that there weren't small victories this week. Mr. Burton quoted Ambassador Jordan as saying that so long as he was America's envoy to Saudi Arabia, no American women seeking refuge would be turned away by the embassy. It too is a welcome reversal: Back in 1990, when a desperate Monica Stowers sought sanctuary for herself and her two children, a U.S. foreign service officer informed her that the embassy was "not a hotel" and ultimately called in the Marine guards to escort the mother and her children out.

A week ago, President Bush's spokesman noted the administration's frustration with the Saudis. We should add the obvious: that delay and obfuscation help the Saudis maintain the dismal status quo.

The failure of the State Department to press these cases and its tiresome repetition that nothing can be done under Saudi law only compounds this, and leaves the American victims with fewer rights and comforts than the Saudi detainees resting in Guantanamo: They at least can send and receive mail. Pat Roush has not even got one photograph.

In the end, the facts on the ground haven't changed much. The Wahhabi contempt for America, which the foreign minister seemed to confirm is official policy in this arena, means that this "ally" of the U.S. is a sanctuary for those who defy American law and American courts.

With Sept. 11 and its many Saudi reminders just around the corner, surely this is a perverse anniversary present to the American people.