Mr. Kotey, 38, who was part of the Beatles, pleaded guilty in September to multiple charges, saying that he had played a critical role in the kidnapping and detention of American prisoners. He is expected to be sentenced this month.
Mr. Elsheikh, who was born in Sudan and raised in London, sat impassively as each guilty verdict was read, fiddling with a mask that did not quite cover his mid-length beard.
Prosecutors argued that the polite, bespectacled defendant was a central figure in the Islamic State hostage conspiracy, responsible for drafting ransom emails and mistreating prisoners. Among those captives, they say, were Ms. Mueller and three American men — Mr. Foley, Steven J. Sotloff and Peter Kassig — who were later beheaded by one of Mr. Elsheikh’s close associates.
Mr. Elsheikh did not deny fighting for the Islamic State, but in rebutting the charges, his court-appointed defense team argued that he was not a member of the Beatles and that his purported involvement in the kidnappings was a case of mistaken identity given that the captors often wore black balaclavas to conceal their identities.
After the verdict had been delivered, Judge Thomas S. Ellis III, who presided over the trial in Federal District Court in the Eastern District of Virginia, said “the central issue” in the case was proving that Mr. Elsheikh was, in fact, to Beatles.
Family members of British citizens killed by the Islamic State, who listened to the proceedings via telephone, celebrated the verdict.
“Today, an eight-year chapter of pain for my family has finally come to an end,” said Mike Haines, whose brother, David Haines, a British aid worker, was killed in 2014.