  We present a search for all embedded protostars with internal luminosities <
1 solar luminosity in the sample of nearby, low-mass star-forming regions
surveyed by the Spitzer Space Telescope c2d Legacy Project. The internal
luminosity (Lint) of a source is the luminosity of the central source and
excludes luminosity arising from external heating. On average, the c2d data are
sensitive to embedded protostars with Lint > 4E-3 (d/140 pc)^2 solar
luminosities, a factor of 25 better than the sensitivity of IRAS to such
objects. We present selection criteria used to identify candidates from the
Spitzer data and examine complementary data to decide whether each candidate is
truly an embedded protostar. We find a tight correlation between the 70 micron
flux and internal luminosity of a protostar, an empirical result based on
observations and two-dimensional radiative transfer models of protostars. We
identify 50 embedded protostars with Lint < 1 solar luminosities; 15 have Lint
< 0.1 solar luminosities. The intrinsic distribution of source luminosities
increases to lower luminosities. While we find sources down to the above
sensitivity limit, indicating that the distribution may extend to luminosities
lower than probed by these observations, we are able to rule out a continued
rise in the distribution below 0.1 solar luminosities. Between 75-85% of cores
classified as starless prior to being observed by Spitzer remain starless to
our luminosity sensitivity; the remaining 15-25% harbor low-luminosity,
embedded protostars. We compile complete Spectral Energy Distributions for all
50 objects and calculate standard evolutionary signatures, and argue that these
objects are inconsistent with the simplest picture of star formation wherein
mass accretes from the core onto the protostar at a constant rate.
