We present new results from a study of recent cosmic microwave background (CMB) observations which provide the first strong evidence for the existence of a 'Doppler peak' localized in both angular scale and amplitude. This first estimate of the angular position of the peak is used to place a new direct limit on the curvature of the universe, corresponding to a density of Omega = 0.7 +0.8, -0.5, consistent with a flat universe. Very low-density 'open' universe models are inconsistent with this limit unless there is a significant contribution from a cosmological constant. For a flat standard cold dark matter dominated universe we use our results in conjunction with big bang nucleosynthesis constraints to determine the value of the Hubble constant as H0 = 30-70 km/s Mpc for baryon fractions Omega(b) = 0.05 to 0.2. For H0 = 50 km/s Mpc we find the primordial spectral index of the fluctuations to be n = 1.1 +/- 0.1, in close agreement with the inflationary prediction of n of about 1.0.