Prosecutors, it turned out, were pursuing a notable user of Lavabit, Mr. Levison’s secure e-mail service: Edward J. Snowden, the former National Security Agency contractor who leaked classified documents that have put the intelligence agency under sharp scrutiny. Mr. Levison was willing to allow investigators with a court order to tap Mr. Snowden’s e-mail account; he had complied with similar narrowly targeted requests involving other customers about two dozen times.

But they wanted more, he said: the passwords, encryption keys and computer code that would essentially allow the government untrammeled access to the protected messages of all his customers. That, he said, was too much.

“You don’t need to bug an entire city to bug one guy’s phone calls,” Mr. Levison, 32, said in a recent interview. “In my case, they wanted to break open the entire box just to get to one connection.”

On Aug. 8, Mr. Levison closed Lavabit rather than, in his view, betray his promise of secure e-mail to his customers.