  We present a three-pointing study of the molecular gas in the starburst
nucleus of M82 based on 190 - 307 GHz spectra obtained with Z-Spec at the
Caltech Submillimeter Observatory. We present intensity measurements,
detections and upper limits, for 20 transitions, including several new
detections of CS, HNC, C2H, H2CO, and CH3CCH lines. We combine our measurements
with previously-published measurements at other frequencies for HCN, HNC, CS,
C34S, and HCO+ in a multi-species likelihood analysis constraining gas mass,
density and temperature, and the species' relative abundances. We find some 1.7
- 2.7 x 10^8 M_sun of gas with n_H2 between 1 - 6 x 10^4 cm^-3 and T > 50 K.
While the mass and temperature are comparable to values inferred from mid-J CO
transitions, the thermal pressure is a factor of 10 - 20 greater. The molecular
interstellar medium is largely fragmented and is subject to ultraviolet
irradiation from the star clusters. It is also likely subject to cosmic rays
and mechanical energy input from the supernovae, and is warmer on average than
the molecular gas in the massive star formation regions in the Milky Way. The
typical conditions in the dense gas in M82's central kpc appear unfavorable for
further star formation; if any appreciable stellar populations are currently
forming, they are likely biased against low mass stars, producing a top-heavy
initial mass function.
