We have used the GALEX (Galaxy Evolution Explorer) spectroscopic survey mode, with a resolution of image8 Aa in the far-ultraviolet (FUV; 1350-1750 Aa) and image20 Aa in the near-ultraviolet (NUV; 1950-2750 Aa) for a systematic search of Ly alpha -emitting galaxies at low redshift. Our aim is to fill a gap between high-redshift surveys and a small set of objects studied in detail in the nearby universe. A blind search of 7018 spectra extracted in five deep exposures (5.65 deg super(2)) has resulted in 96 Ly alpha - emitting galaxy candidates in the FUV domain after accounting for broad-line AGNs. The Ly alpha equivalent widths (EWs) are consistent with stellar population model predictions and show no trends as a function of UV color or UV luminosity, with the exception of a possible decrease in the most luminous objects that may be due to small-number statistics. The objects' distribution in EW is similar to that at image, but their fraction among star-forming galaxies is smaller. Avoiding uncertain candidates, a subsample of 66 objects in the range image has been used to build a Ly alpha luminosity function (LF). The incompleteness due to objects with significant Ly alpha emission but a UV continuum too low for spectral extraction has been evaluated. A comparison with H alpha LFs in the same redshift domain is consistent with an average Ly alpha /H alpha of image1 in about 15% of the star-forming galaxies. A comparison with high-redshift Ly alpha LFs implies an increase of the Ly alpha luminosity density by a factor of about 16 from image to image. By comparison with the factor of 5 increase in the UV luminosity density in the same redshift range, this suggests an increase of the average Ly alpha escape fraction with redshift.