We present a determination by the Archeops experiment of the angular power spectrum of the cosmic microwave background anisotropy in 16 bins over the multipole range ℓ = 15 - 350. Archeops was conceived as a precursor of the Planck HFI instrument by using the same optical design and the same technology for the detectors and their cooling. Archeops is a balloon-borne instrument consisting of a 1.5 m aperture diameter telescope and an array of 21 photometers maintained at ~100 mK that are operating in 4 frequency bands centered at 143, 217, 353 and 545 GHz. The data were taken during the Arctic night of February 7, 2002 after the instrument was launched by CNES from Esrange base (Sweden). The entire data cover ~30% of the sky. This first analysis was obtained with a small subset of the dataset using the most sensitive photometer in each CMB band (143 and 217 GHz) and 12.6% of the sky at galactic latitudes above 30 degrees where the foreground contamination is measured to be negligible. The large sky coverage and medium resolution (better than 15 arcmin) provide for the first time a high signal-to-noise ratio determination of the power spectrum over angular scales that include both the first acoustic peak and scales probed by COBE/DMR. With a binning of Δ ℓ = 7 to 25 the error bars are dominated by sample variance for ℓ below 200. A companion paper details the cosmological implications.