Mr. Goldstein waited a week, went online again and was still able to gain access to the Florida woman’s account. The Goldsteins then spoke to their fifth Amex customer rep, a man who, Mr. Goldstein said, did seem to grasp the seriousness. “He said this could be criminal,” Mr. Goldstein recalled. “We said, ‘Exactly.’ ” When the Goldsteins commended him for his efforts, he asked if they would repeat that to his supervisor. “In these times, maybe he was trying to hold on to his job,” Mr. Goldstein said.

The supervisor, their sixth Amex rep, said she could see that the technical people were all over it.

The Goldsteins went on vacation to St. Croix, returned Jan. 8 and, on Jan. 9, Mr. Goldstein could still hack into the account. “I could see all the shopping she’d done while we were gone,” he said. “I also saw she didn’t pay her last balance and it was a lot — over $4,000.”

He wondered whether anyone had told this woman what had happened.

That weekend, Mr. Goldstein laid out the saga for me (he said we had met years ago, though I had no memory of it) and on Monday morning — exactly three weeks after the Goldsteins first tried to turn themselves in — I called Rosa Alfonso, an Amex spokeswoman, who immediately grasped the seriousness. “This is all I’m working on,” she said.

I explained that Mr. Goldstein was eager to give her the information for the hacked account. But Ms. Alfonso said the privacy issues were so sensitive that she couldn’t take it herself and needed to work through a special executive customer service rep, named Ed.

“I told Ed the entire story,” Mr. Goldstein later reported to me. “Complete silence. I don’t know if he was stunned or upset I was taking up so much of his time.” Mr. Goldstein was surprised how little information Ed seemed to need from him.