  The two point angular correlation function is an excellent measure of
structure in the universe. To extract from it the three dimensional power
spectrum, one must invert Limber's Equation. Here we perform this inversion
using a Bayesian prior constraining the smoothness of the power spectrum. Among
other virtues, this technique allows for the possibility that the estimates of
the angular correlation function are correlated from bin to bin. The output of
this technique are estimators for the binned power spectrum and a full
covariance matrix. Angular correlations mix small and large scales but after
the inversion, small scale data can be trivially eliminated, thereby allowing
for realistic constraints on theories of large scale structure. We analyze the
APM catalogue as an example, comparing our results with previous results. As a
byproduct of these tests, we find -- in rough agreement with previous work --
that APM places stringent constraints on Cold Dark Matter inspired models, with
the shape parameter constrained to be $0.25\pm 0.04$ (using data with
wavenumber $k \le 0.1 h{\rm Mpc}^{-1}$). This range of allowed values use the
full power spectrum covariance matrix, but assumes negligible covariance in the
off-diagonal angular correlation error matrix, which is estimated with a large
angular resolution of 0.5degrees (in the range 0.5 and 20 degrees).
