  We present the first sample of 31-GHz selected sources to flux levels of 1
mJy. From late 2005 to mid 2007, the Sunyaev-Zel'dovich Array (SZA) observed
7.7 square degrees of the sky at 31 GHz to a median rms of 0.18 mJy/beam. We
identify 209 sources at greater than 5 sigma significance in the 31 GHz maps,
ranging in flux from 0.7 mJy to ~200 mJy. Archival NVSS data at 1.4 GHz and
observations at 5 GHz with the Very Large Array are used to characterize the
sources. We determine the maximum-likelihood integrated source count to be
N(>S) = (27.2 +- 2.5) deg^-2 x (S_mJy)^(-1.18 +- 0.12) over the flux range 0.7
- 15 mJy. This result is significantly higher than predictions based on 1.4-GHz
selected samples, a discrepancy which can be explained by a small shift in the
spectral index distribution for faint 1.4-GHz sources. From comparison with
previous measurements of sources within the central arcminute of massive
clusters, we derive an overdensity of 6.8 +- 4.4, relative to field sources.
