Smith, 26, already leads the one-day side and will also captain the Twenty20 team in the absence of the injured Aaron Finch.
Fellow batsman David Warner, 28, has been named as Smith's vice-captain.
"At 26, Steve is a fine young man with extraordinary talent," said national selector Rod Marsh.
"He is highly regarded by the selectors and we congratulate him on being appointed to the role on an ongoing basis. He should be incredibly proud."
Clarke, 34, confirmed his decision to retire after his side surrendered the Ashes with a heavy defeat by England at Trent Bridge, which gave the hosts a 3-1 lead.
Smith had already captained the side in three Tests against India last summer, when Clarke was recovering from hamstring surgery and back issues.
The New South Welshman has played in 32 Tests for his country, scoring 2,952 runs at an average of 54.66.
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"When Michael made his decision to retire last week it was a very straightforward decision for us to nominate Steve as his successor," added Marsh.
"He has big shoes to fill but everything about him suggests he is the right man for the job."
Warner has earned the vice-captain's role despite a history of disciplinary problems, and Marsh said of the opener: "David has matured and developed into an important senior figure in the Australian team. He has come a long way.
"We believe that he will respond well to the added responsibility of leadership."
Smith and Warner will lead Australia in the limited-overs matches that follow the final Ashes Test at the Oval, which starts on 20 August.
Steve Smith will become Australia's Test captain after the Ashes following Michael Clarke's decision to retire at the end of the series.