In the course of our ongoing search for serendipitous high-redshift Ly alpha emission in deep archival Keck spectra, we discovered two very high equivalent width (W-lambda(obs) greater than or similar to 450 Angstrom; 2 sigma) Ly alpha emission-line candidates at z similar to 3 in a moderate dispersion (lambda/Delta lambda similar or equal to 1200) spectrogram Both lines have low velocity dispersions (sigma(v) similar to 60 km s(-1)) and deconvolved radii r approximate to 1 h(50)(-1) kpc. We argue that the lines are Ly alpha and are powered by stellar ionization. The surface density of robust, high equivalent width Ly alpha candidates is estimated to be similar to 3 +/- 2 arcmin(-2) per unit redshift at z similar or equal to 3, consistent with the recent estimate of Cowie & Hu. The Ly alpha emission-line source characteristics are consistent with the galaxies undergoing their first burst of star formation, i.e., with being primeval. Source sizes and velocity dispersions are comparable to the theoretical primeval galaxy model of Lin & Murray based on the inside out, self-similar collapse of an isothermal sphere. In this model, star formation among field galaxies is a protracted process. Galaxies are thought to be able to display high equivalent widths for only about the first few times 10(7) yr. This time is short in relation to the difference in look-back times between z = 3 and z = 4 and implies that a substantial fraction of strong line-emitting galaxies at z = 3 were formed at redshifts z less than or equal to 4. We discuss the significance of high equivalent width Ly alpha-emitting galaxies in terms of the emerging picture of the environment and the specific characteristics of primeval galaxy formation at high redshift.