The spin-1/2 kagome antiferromagnet is an archetypal frustrated system predicted to host a variety of exotic magnetic states. We show using neutron scattering measurements that deuterated vesignieite BaCu_{3}V_{2}O_{8}(OD)_{2}, a fully stoichiometric S=1/2 kagome magnet with <1% lattice distortion, orders magnetically at T_{N}=9 K into a multi-k coplanar variant of the predicted triple-k octahedral structure. We find that this structure is stabilized by a dominant antiferromagnetic third-neighbor exchange J_{3} with minor first- or second-neighbor exchanges. The spin-wave spectrum is well described by a J_{3}-only model including a tiny symmetric exchange anisotropy.