We have searched for the decay B+-->omegal(+)nu (l=e or mu) in 78 fb(-1) of Upsilon(4S) data (85x10(6)BB events) accumulated with the Belle detector. The final state is fully reconstructed using the omega decay into pi(+)pi(-)pi(0), combined with detector hermeticity to estimate the neutrino momentum. A signal of 414+/-125 events is found in the data, corresponding to a branching fraction of (1.3+/-0.4+/-0.2+/-0.3)x10(-4), where the first two errors are statistical and systematic, respectively. The third error reflects the estimated form-factor uncertainty.