Anthony Joseph admitted manslaughter on grounds of diminished responsibility after a murder trial jury was discharged for the second time.
Joseph had been released from prison earlier in the day despite an outstanding arrest warrant for him.
Richard Whelan, 28, was stabbed seven times on a bus in Holloway Road, north London, in July 2005.
Josephs legal team claimed he was suffering from paranoid schizophrenia.
Mr Whelan's family said they were very disappointed with the vedict Joseph will be sentenced on 20 December at the Old Bailey.
A statement released on behalf of Mr Whelan's family immediately after the verdict said: "Having heard the whole of the evidence for the second time we feel that the defence of diminished responsibility in this case has been used as a defence for the un-defendable, with so much evidence showing that Anthony Joseph was an angry and vindictive man."
"He has tried to excuse his actions that evening by claiming mental illness, however in our opinion he callously killed Richard for no reason at all."
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