Starting in 2013 February, Swift has been performing short daily monitoring observations of the G2 gas cloud near Sgr A* with the X-Ray Telescope to determine whether the cloud interaction leads to an increase in the flux from the Galactic center. On 2013 April 24 Swift detected an order of magnitude rise in the X-ray flux from the region near Sgr A*. Initially thought to be a flare from Sgr A*, detection of a short hard X-ray burst from the same region by the Burst Alert Telescope suggested that the flare was from an unresolved new Soft Gamma Repeater, SGR J1745-29. Here we present the discovery of SGR J1745-29 by Swift, including analysis of data before, during, and after the burst. We find that the spectrum in the 0.3-10 keV range is well fit by an absorbed blackbody model with kTBB ~ 1 keV and absorption consistent with previously measured values from the quiescent emission from Sgr A*, strongly suggesting that this source is at a similar distance. Only one SGR burst has been detected so far from the new source, and the persistent light curve shows little evidence of decay in approximately 2 weeks of monitoring after outburst. We discuss this light curve trend and compare it with those of other well covered SGR outbursts. We suggest that SGR J1745-29 belongs to an emerging subclass of magnetars characterized by low burst rates and prolonged steady X-ray emission 1-2 weeks after outburst onset.