We give recursive formulae for the exact removal of the contribution of the center-of-mass spurious states from the fixed-spin and parity nuclear level density found in shell-model calculations, provided the total level density for restricted configurations is known. The method is valid for a large class of problems using a harmonic oscillator basis. Using our earlier methods based on statistical spectroscopy that utilize the centroids and widths for a restricted class of fixed-spin configurations, such as Nvariant Planck's over 2piomega excitations, one can calculate very accurately level densities free of spurious states. The approach is applicable to other fermion and boson systems trapped by an oscillator potential.