  Spatially varying primordial magnetic fields may be efficiently dissipated
prior to the epoch of recombination due to the large viscosity of the
baryon-photon fluid. We show that this dissipation may result in observable
chemical potential mu and Compton y distortions in the cosmic microwave
background radiation (CMB) spectrum. Current upper limits on mu and y from
FIRAS constrain magnetic fields to have strength B_0 < 3\times 10^{-8}Gauss
(scaled to the present) between comoving coherence length \approx 400 pc and
\approx 0.6 Mpc. This represents the strongest upper limit on small-scale
primordial magnetic fields to date.
