We report the Fermi Large Area Telescope discovery of gamma-ray pulsations from the 22.7 ms pulsar A in the double pulsar system J0737-3039A/B. This is the first mildly recycled millisecond pulsar (MSP) detected in the GeV domain. The 2.7 s companion object PSR J0737-3039B is not detected in gamma rays. PSR J0737-3039A is a faint gamma-ray emitter, so that its spectral properties are only weakly constrained; however, its measured efficiency is typical of other MSPs. The two peaks of the gamma-ray light curve are separated by roughly half a rotation and are well offset from the radio and X-ray emission, suggesting that the GeV radiation originates in a distinct part of the magnetosphere from the other types of emission. From the modeling of the radio and the gamma-ray emission profiles and the analysis of radio polarization data, we constrain the magnetic inclination $alpha$ and the viewing angle $zeta$ to be close to 90$^circ$, which is consistent with independent studies of the radio emission from PSR J0737-3039A. A small misalignment angle between the pulsars spin axis and the systems orbital axis is therefore favored, supporting the hypothesis that pulsar B was formed in a nearly symmetric supernova explosion as has been discussed in the literature already.
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