  We investigate the antiferromagnetic canting instability of the spin-1/2
kagome ferromagnet, as realized in the layered cuprates
Cu$_3$Bi(SeO$_3)_2$O$_2$X (X=Br, Cl, and I). While the local canting can be
explained in terms of competing exchange interactions, the direction of the
ferrimagnetic order parameter fluctuates strongly even at short distances on
account of frustration which gives rise to an infinite ground state degeneracy
at the classical level. In analogy with the kagome antiferromagnet, the
accidental degeneracy is fully lifted only by non-linear 1/S corrections,
rendering the selected uniform canted phase very fragile even for spins-1/2, as
shown explicitly by coupled-cluster calculations. To account for the observed
ordering, we show that the minimal description of these systems must include
the microscopic Dzyaloshinsky-Moriya interactions, which we obtain from
density-functional band-structure calculations. The model explains all
qualitative properties of the kagome francisites, including the detailed nature
of the ground state and the anisotropic response under a magnetic field. The
predicted magnon excitation spectrum and quantitative features of the
magnetization process call for further experimental investigations of these
compounds.
