  Primordial magnetic fields will generate non-Gaussian signals in the cosmic
microwave background (CMB) as magnetic stresses and the temperature anisotropy
they induce depend quadratically on the magnetic field. We compute a new
measure of magnetic non-Gaussianity, the CMB trispectrum, on large angular
scales, sourced via the Sachs-Wolfe effect. The trispectra induced by magnetic
energy density and by magnetic scalar anisotropic stress are found to have
typical magnitudes of approximately a few times 10^{-29} and 10^{-19},
respectively. Observational limits on CMB non-Gaussianity from WMAP data allow
us to conservatively set upper limits of a nG, and plausibly sub-nG, on the
present value of the primordial cosmic magnetic field. This represents the
tightest limit so far on the strength of primordial magnetic fields, on Mpc
scales, and is better than limits from the CMB bispectrum and all modes in the
CMB power spectrum. Thus, the CMB trispectrum is a new and more sensitive probe
of primordial magnetic fields on large scales.
