The 69-year-old Justin Welby had already announced his resignation in November after the scandal came to light, but initially continued in office. His term of office ended at midnight after around twelve years.
The Archbishop of York, Stephen Cottrell, is to take over Welby's duties temporarily.
It could be months before a successor is chosen.
The selection process is in the hands of the Crown Nominations Commission, a committee consisting of 16 voting members.
It will select one top candidate and a second who would also be suitable.
The former head of the British domestic intelligence service MI5, Jonathan Douglas Evans, was appointed head of the commission in mid-December.
Welby had announced in November that he would be stepping down. He said he bore personal and institutional responsibility for mistakes made in recent years.
He had previously been heavily implicated by a new investigation report.
According to the report, Welby had known about the decades of abuse by a helper in church youth camps since 2013.
Nevertheless, nothing was done to help clear up the matter.