Power Spectrum Constraints from Spectral Distortions in the Cosmic Microwave Background

%The content of this replacement paper is identical to the original. %We have attempted to fix the postscript so that it will print out on %a larger number of printers. Using recent experimental limits on $\mu$ distortions from COBE FIRAS, and the large lever-arm spanning the damping of sub-Jeans scale fluctuations to the scale of the COBE DMR fluctuations, we set a constraint on the slope of the primordial power spectrum $n$. It is possible to analytically calculate the contribution over the full range of scales and redshifts, correctly taking into account fluctuation growth and damping as well as thermalization processes. We find that the 95\% upper limit is weakly dependent on cosmological parameters, e.g. $n<1.54 (h=0.5)$ and $n<1.56 (h=1.0)$ for $\Omega_0=1$ with marginally weaker constraints for $\Omega_0<1$ in a flat $\Omega_0 +\Omega_\Lambda=1$ universe.