We set new constraints on a seven-dimensional space of cosmological parameters within the class of inflationary adiabatic models . We use the angular power spectrum of the cosmic microwave background measured over a wide range of \ell in the first flight of the MAXIMA balloon-borne experiment ( MAXIMA-1 ) and the low \ell results from COBE/DMR . We find constraints on the total energy density of the universe , \Omega = 1.0 ^ { +0.15 } _ { -0.30 } , the physical density of baryons , \Omega _ { b } h ^ { 2 } = 0.03 \pm 0.01 , the physical density of cold dark matter , \Omega _ { cdm } h ^ { 2 } = 0.2 ^ { +0.2 } _ { -0.1 } , and the spectral index of primordial scalar fluctuations , n _ { s } = 1.08 \pm 0.1 , all at the 95 \% confidence level . By combining our results with measurements of high-redshift supernovae we constrain the value of the cosmological constant and the fractional amount of pressureless matter in the universe to 0.45 < \Omega _ { \Lambda } < 0.75 and 0.25 < \Omega _ { m } < 0.50 , at the 95 \% confidence level . Our results are consistent with a flat universe and the shape parameter deduced from large scale structure , and in marginal agreement with the baryon density from big bang nucleosynthesis .