  We have searched for the decay $B^+\to\omega l^+\nu$ ($l=e$ or $\mu$) in 78
fb$^{-1}$ of $\Upsilon(4S)$ data (85 million $B\bar B$ 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 $414\pm 125$ events is found in the data,
corresponding to a branching fraction of $(1.3\pm 0.4\pm 0.2\pm 0.3)\times
10^{-4}$, where the first two errors are statistical and systematic,
respectively. The third error reflects the estimated form-factor uncertainty.
