We report the first detections of the Class 0 protostellar source IRAM 04191+1522 at wavelengths shortward of 60 microns with the Spitzer Space Telescope. We see extended emission in the Spitzer images that suggests the presence of an outflow cavity in the circumstellar envelope. We combine the Spitzer observations with existing data to form a complete dataset ranging from 3.6 to 1300 microns and use these data to construct radiative transfer models of the source. We conclude that the internal luminosity of IRAM 04191+1522, defined to be the sum of the luminosity from the internal sources (a star and a disk), is L_int = 0.08 +/- 0.04 L_sun, placing it among the lowest luminosity protostars known. Though it was discovered before the launch of the Spitzer Space Telescope, IRAM 04191+1522 falls within a new class of Very Low Luminosity Objects being discovered by Spitzer. Unlike the two other well-studied objects in this class, which are associated either with weak, compact outflows or no outflows at all, IRAM 04191+1522 has a well-defined molecular outflow with properties consistent with those expected based on relations derived from higher luminosity (L_int > 1 L_sun) protostars. We discuss the difficulties in understanding IRAM 04191+1522 in the context of the standard model of star formation, and suggest a possible explanation for the very low luminosity of this source.