Councillor Peach, 46, was elected to the city council's Park Ward in May 1997 when Peterborough was awarded unitary authority status. He previously represented Park Ward on Cambridgeshire County Council for nine years and was a member of the Cambridgeshire Police Authority and Cambridgeshire and Peterborough Fire Authority.
He said: 'Obviously, I have mixed emotions about taking over as leader under such tragic circumstances. Councillor Franklin was an energetic advocate for Peterborough's growth potential and I want to build on the excellent initiatives that he championed.
'The city council and its partners are committed to achieving some impressive targets as we aim to build a better Peterborough for the benefit of all residents. I will always be batting for our city and its residents.'
A service for the life of the late Councillor Franklin will be held at Peterborough Cathedral on 19 April.