Based on the change-of-coordinates that was previously proposed in [1], in this paper, the wave spectrum inversion problem is solved with very high accuracy for the case when there is little or no noise in the second-order Doppler spectrum. The new solution is based on the form of the forward problem, which is a sum of spatially-dependent linear convolution and cross-correlations. This sum can be solved precisely by solving for the wave spectrum power spectral densities iteratively from larger to smaller frequencies, and it is shown here on synthetic data that the solution is exact up to numerical error in the absence of noise. This exact solution is found to be sensitive to noise in the Doppler spectrum, and thus a second algorithm is derived which reduces this sensitivity and still arrives at a sensible solution for the nondirectional ocean wave spectrum.