  The spin-$\frac{1}{2}$ 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
$T_{\mathrm{N}}=9$K into a multi-k coplanar variant of the predicted triple-k
octahedral structure. We find this structure is stabilized by a dominant
antiferromagnetic 3$^{\mathrm{rd}}$-neighbor exchange $J_3$ with minor
1$^{\mathrm{st}}$- or 2$^{\mathrm{nd}}$--neighbour exchange. The spin-wave
spectrum is well described by a $J_3$-only model including a tiny symmetric
exchange anisotropy.
