  We measure the angular power spectrum of the WMAP first-year temperature
anisotropy maps. We use SpICE (Spatially Inhomogeneous Correlation Estimator)
to estimate Cl's for multipoles l=2-900 from all possible cross-correlation
channels. Except for the map-making stage, our measurements provide an
independent analysis of that by Hinshaw etal (2003). Despite the different
methods used, there is virtually no difference between the two measurements for
l < 700 ; the highest l's are still compatible within 1-sigma errors. We use a
novel intra-bin variance method to constrain Cl errors in a model independent
way. When applied to WMAP data, the intra-bin variance estimator yields
diagonal errors 10% larger than those reported by the WMAP team for 100 < l <
450. This translates into a 2.4 sigma detection of systematics since no
difference is expected between the SpICE and the WMAP team estimator window
functions in this multipole range. With our measurement of the Cl's and errors,
we get chi^2/d.o.f. = 1.042 for a best-fit LCDM model, which has a 14%
probability, whereas the WMAP team obtained chi^2/d.o.f. = 1.066, which has a
5% probability. We assess the impact of our results on cosmological parameters
using Markov Chain Monte Carlo simulations. From WMAP data alone, assuming
spatially flat power law LCDM models, we obtain the reionization optical depth
tau = 0.145 +/- 0.067, spectral index n_s = 0.99 +/- 0.04, Hubble constant h =
0.67 +/- 0.05, baryon density Omega_b h^2 = 0.0218 +/- 0.0014, cold dark matter
density Omega_{cdm} h^2 = 0.122 +/- 0.018, and sigma_8 = 0.92 +/- 0.12,
consistent with a reionization redshift z_{re} = 16 +/- 5 (68% CL).
