We study the stochastic multi-armed bandit problem when one knows the value μ(⋆) of an optimal arm, as a well as a positive lower bound on the smallest positive gap Δ. We propose a new randomized policy that attains a regret uniformly bounded over time in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows Δ, and bounded regret of order 1/Δ is not possible if one only knows μ(⋆).