American doctor sprang up to give name (7)
I believe the answer is:
'give name' is the definition.
The definition suggests a singular noun which matches the answer.
'american doctor sprang up' is the wordplay.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'doctor' becomes 'MB' (Bachelor of Medicine, Medicinae Baccalaureus).
'sprang up' becomes 'rose' (rise can mean to spring up).
'to' acts as a link.
Can you help me to learn more?
(Other definitions for ambrose that I've seen before include "Bishop-saint" , "Church Father" , "-- Bierce, US author" , "name" , "Bishop of Milan once" .)