His only rival for the position, Conservative leader James Kerr got just two votes – his own and fellow Conservative Sarah Patrick.
All four members of the SNP on the board abstained from the vote and did not put a candidate forward, leaving the way clear for Provost Bissett to be elected.
However, SNP Councillor Bryan Deakin was elected unopposed as the depute convener of the board.
The board met on Wednesday for the first time since the local government elections were held on May 5.
There are several new faces among the members.
The full list of members is now: Councillor David Balfour (SNP); Councillor Lorna Binnie (SNP); Provost Robert Bissett (Lab); Councillor William Buchanan (Ind); Councillor Bryan Deakin (SNP); Councillor Alf Kelly (Lab); Councillor James Kerr (Con); Councillor Sarah Patrick (Con); Councillor James Robertson (SNP); Councillor Euan Stainbank (Lab).