From 1985 to 1986 he was director and head of the Swedish Church Board for worship and evangelism.

He then became bishop of the Diocese of Luleå and in 1993 Archbishop of Uppsala and Primate of Sweden.

Weman was Secretary of the National Association of Youth and Church of Sweden Christian student movement and a member of the Swedish church's mission board and its executive committee.

In 2006 he became doctor of theology with a thesis entitled Contemporary worship and medieval churches.

[2] Weman died in Sigtuna at the age of 91 on 24 February 2024, the day before he would have turned 92.