  We studied the radiative muon decay $\mu^+ \to e^+\nu\bar{\nu}\gamma$ by
using for the first time an almost fully polarized muon source. We identified a
large sample (~13000) of these decays in a total sample of 1.8x10^14 positive
muon decays collected in the MEG experiment in the years 2009--2010 and
measured the branching ratio B($\mu^+ \to e^+\nu\bar{\nu}\gamma$) =
(6.03+-0.14(stat.)+-0.53(sys.))x10^-8 for E_e > 45 MeV and E_{\gamma} > 40 MeV,
consistent with the Standard Model prediction. The precise measurement of this
decay mode provides a basic tool for the timing calibration, a normalization
channel, and a strong quality check of the complete MEG experiment in the
search for $\mu^+ \to e^+\gamma$ process.
