We present a map and an angular power spectrum of the anisotropy of the cosmic microwave background (CMB) from the first flight of the Millimeter-wave Anisotropy Experiment Imaging Array (MAXIMA). MAXIMA is a balloon-borne experiment with an array of 16 bolometric photometers operated at 100 mK. MAXIMA observed a 124 deg2 region of the sky with 10' resolution at frequencies of 150, 240, and 410 GHz. The data were calibrated using in-flight measurements of the CMB dipole anisotropy. A map of the CMB anisotropy was produced from three 150 and one 240 GHz photometer without need for foreground subtractions. Analysis of this CMB map yields a power spectrum for the CMB anisotropy over the range 36<=l<=785. The spectrum shows a peak with an amplitude of 78+/-6 μK at l~=220 and an amplitude varying between ~40 and ~50 μK for 400<~l<~785.