A study of W+W- events accomanied by hard photon radiation produced in e+e- collisions at LEP is presented. Events consistent with being two on-shell W bosons and an isolated photon are selected from 681 pb^-1 of data recorded at 180 GeV < sqrt(s) < 209 GeV. For these data , 187 W+W- candidates are selected with photon energies greater than 2.5 GeV. The selected events are used to determine the W+ W- gamma cross section at five values of sqrt(s). The results are consistent with the Standard Model expectation. These data provide constraints on the related O(alpha) systematic uncertainties on the measurement of the W boson mass at LEP. Finally, the data are used to derive 95% C.L. upper limits on possible anomalous contributions to the W+ W- gamma gamma and W+ W- Z0 gamma vertices.