  We derive model-independent, "naturalness" upper bounds on the magnetic
moments \mu_\nu of Dirac neutrinos generated by physics above the scale of
electroweak symmetry breaking. In the absence of fine-tuning of effective
operator coefficients, we find that current information on neutrino mass
implies that |\mu_\nu | < 10^(-14) Bohr magnetons. This bound is several orders
of magnitude stronger than those obtained from analyses of solar and reactor
neutrino data and astrophysical observations.
