  We have performed a search for the rare leptonic decay B+ --> mu+ nu_mu with
data collected at the Upsilon(4S) resonance by the BABAR experiment at the
PEP-II storage ring. In a sample of 88.4 million BBbar pairs, we find no
significant evidence for a signal and set an upper limit on the branching
fraction BR(B+ --> mu+ nu_mu) < 6.6 x 10^{-6} at the 90% confidence level.
