We describe the results of an extremely deep, 0.28 deg 2 survey for z= 3.1 Lyα emission-line galaxies in the Extended Chandra Deep Field-South. By using a narrowband 5000 Å filter and complementary broadband photometry from the MUSYC survey, we identify a statistically complete sample of 162 galaxies with monochromatic fluxes brighter than 1.5× 10-17 ergs cm-2 s-1 and observer's frame equivalent widths greater than 80 Å. We show that the equivalent width distribution of these objects follows an exponential with a rest-frame scale length of w 0= 76 Å. In addition, we show that in the emission line, the luminosity function of Lyα galaxies has a faint-end power-law slope of α=-1.49, a bright-end cutoff of log L*= 42.64, and a space density above our detection thresholds of (1.46±0.12)× 10-3 h galaxies Mpc-3. Finally, by comparing the emission-line and continuum properties of the Lyα emitters, we show that the star formation rates derived from Lyα are~ 3 times lower than those inferred from the rest-frame UV continuum. We use this offset to deduce the existence of a small amount of internal extinction within the host galaxies. This extinction, coupled with the lack of extremely high equivalent width emitters, argues that these galaxies are not primordial Population III objects, although they are young and relatively chemically unevolved.