  We present Green Bank Telescope (GBT) observations of the 3(12)-3(13) (29
GHz) and 4(13)-4(14) (48 GHz) transitions of the H2CO molecule toward a sample
of 23 well-studied star-forming regions. Analysis of the relative intensities
of these transitions can be used to reliably measure the densities of molecular
cores. Adopting kinetic temperatures from the literature, we have employed a
Large Velocity Gradient (LVG) model to derive the average hydrogen number
density [n(H2)] within a 16 arcsecond beam toward each source. Densities in the
range of 10^{5.5}--10^{6.5} cm^{-3} and ortho-formaldehyde column densities per
unit line width between 10^{13.5} and 10^{14.5} cm^{-2} (km s^{-1})^{-1} are
found for most objects, in general agreement with existing measurements. A
detailed analysis of the advantages and limitations to this densitometry
technique is also presented. We find that H2CO 3(12)-3(13)/4(13)-4(14)
densitometry proves to be best suited to objects with T_K >~ 100 K, above which
the H2CO LVG models become relatively independent of kinetic temperature. This
study represents the first detection of these H2CO K-doublet transitions in all
but one object in our sample. The ease with which these transitions were
detected, coupled with their unique sensitivity to spatial density, make them
excellent monitors of density in molecular clouds for future experiments. We
also report the detection of the 9_2--8_1 A^- (29 GHz) transition of CH3OH
toward 6 sources.
