We present measurements of anisotropy in the Cosmic Microwave Background ( CMB ) from the first season of observations with the Degree Angular Scale Interferometer ( DASI ) . The instrument was deployed at the South Pole in the austral summer 1999–2000 , and made observations throughout the following austral winter . We present a measurement of the CMB angular power spectrum in the range 100 < l < 900 in nine bands with fractional uncertainties in the range 10–20 % and dominated by sample variance . In this paper we review the formalism used in the analysis , in particular the use of constraint matrices to project out contaminants such as ground and point source signals and to test for correlations with diffuse foreground templates . We find no evidence of foregrounds other than point sources in the data , and find a maximum likelihood temperature spectral index \beta = -0.1 \pm 0.2 ( 1 \sigma ) , consistent with CMB . We detect a first peak in the power spectrum at l \sim 200 , in agreement with previous experiments . In addition , we detect a peak in the power spectrum at l \sim 550 and power of similar magnitude at l \sim 800 which are consistent with the second and third harmonic peaks predicted by adiabatic inflationary cosmological models .