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A stochastic magnetic field in the early Universe will produce anisotropiesin the temperature and polarization of the cosmic microwave background. Wederive analytic expressions for the microwave background temperature andpolarization power spectra induced by vector and tensor perturbations from apower-law magnetic field. For a scale-invariant stochastic magnetic fieldsmoothed over a comoving scale of $1 {\rm Mpc}$, the MAP satellite has thepotential to constrain the comoving mean-field amplitude to be no greater thanapproximately $2\times10^{-9}$ G. Limits improve as the power-law slopeincreases: for causally-generated power-law magnetic fields, the comovingmean-field amplitude has an upper bound of approximately $4\times10^{-13} {\rmG}$. Such constraints will surpass all current limits on galactic-scaleprimordial stochastic magnetic fields at decoupling.Comment: 28 pages, 3 Postscript figures. Improved treatments on vorticity and  magnetic damping. Revised version in response to the referee's comment

Microwave background signatures of a primordial stochastic magnetic field