On 2009 June 5, the Gamma-ray Burst Monitor (GBM) onboard the Fermi Gamma-ray Space Telescope triggered on two short and relatively dim bursts with spectral properties similar to soft gamma repeater (SGR) bursts. Independent localizations of the bursts by triangulation with the Konus-RF and with the Swift satellite confirmed their origin from the same, previously unknown, source. The subsequent discovery of X-ray pulsations with the Rossi X-ray Timing Explorer confirmed the magnetar nature of the new source, SGR J0418 + 5729. We describe here the Fermi/GBM observations, the discovery and the localization of this new SGR, and our infrared and Chandra X-ray observations. We also present a detailed temporal and spectral study of the two GBM bursts. SGR J0418 + 5729 is the second source discovered in the same region of the sky in the last year, the other one being SGR J0501 + 4516. Both sources lie in the direction of the galactic anti-center and presumably at the nearby distance of ~2 kpc (assuming they reside in the Perseus arm of our Galaxy). The near-threshold GBM detection of bursts from SGR J0418 + 5729 suggests that there may be more such "dim" SGRs throughout our Galaxy, possibly exceeding the population of "bright" SGRs. Finally, using sample statistics, we conclude that the number of observable active magnetars in our Galaxy at any given time is 10, in agreement with our earlier estimates.