AbstractWe use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that, for every odd primep, each side of Darmon's conjectured formula (indexed by positive integersn) is 'almost' ap-adic Kolyvagin system asnvaries. Using the fact that the space of Kolyvagin systems is free of rank one overp, we show that Darmon's formula for arbitrarynfollows from the casen=1, which in turn follows from classical formulas.