We present submillimeter photometry and high spatial resolution 800-mu-m and 450-mu-m maps of the cold, low-luminosity source IRAS 4 (T(d) = 33 K, L approximately 28 L.) in the active star formation region NGC 1333. IRAS 4 is resolved into a binary system, with components separated by approximately 31" and connected by a bridge of faint dust emission. The submillimeter flux density from the two components is unusually bright [S4A(800-mu-m) approximately 12.7 Jy; S4B(800-mu-m) approximately 8.6 Jy]; IRAS 4A is marginally resolved into a disklike structure 3200 x 2200 AU in size. The spectral energy distribution of 4A is best explained by optically thick dust emission shortward of 1.1 mm. We derive a total gas and dust mass of 9.2 M. for 4A and 4.0 M. for 4B. These correspond to average gas densities of 9.0 x 10(9) cm-3 and 9.2 x 10(8) cm-3, respectively, and imply a line-of-sight visual extinction of approximately 7 x 10(4) mag and approximately 1 x 10(4) mag for 4A and 4B, respectively. The dust masses associated with the binary components are by far the largest seen toward any star of comparable luminosity and are a factor of 10 higher than predicted by theoretical models of protostars. We propose that we are seeing an extremely young stellar system, possibly a true protostar, in which most of the luminosity is the result of the accretion process.