  Using a sample of 92 UV continuum-selected, spectroscopically identified
galaxies with <z> = 2.65, all of which have been imaged in the Ly-a line with
extremely deep narrow-band imaging, we examine galaxy Ly-a emission profiles to
very faint surface brightness limits. The galaxies are representative of
spectroscopic samples of LBGs at similar redshifts in terms of apparent
magnitude, UV luminosity, inferred extinction, and star formation rate, and
were selected without regard to Ly-a emission properties. We use extremely deep
stacks of UV continuum and Ly-a emission line images to show that all
sub-samples exhibit diffuse Ly-a emission to radii of at least 10" (80 physical
kpc), including galaxies whose spectra exhibit Ly-a in net absorption. The
intensity scaling, but not the surface brightness distribution, is strongly
correlated with the emission observed in the central ~1". The characteristic
scale length for Ly-a line emission exceeds that of the UV continuum light for
the same galaxies by factors of at least 5-10, regardless of the spectral
morphology of Ly-a. Including the extended Ly-a halos increases the total Ly-a
flux [and rest equivalent width W_0(Lya)] by an average factor of 5. We argue
that most, if not all, of the observed Ly-a emission in the diffuse halos
originates in the galaxy H II regions and is scattered in our direction by H I
gas in the galaxy's circum-galactic medium (CGM). We show that whether or not a
galaxy is classified as a giant "Lyman-a Blob" (LAB) depends sensitively on the
Ly-a surface brightness threshold reached by an observation. Accounting for
diffuse Ly-a halos, all LBGs would be LABs if surveys were routinely sensitive
to 10 times lower surface brightness thresholds; also, essentially all LBGs
would qualify as LAEs (W_0(Lya) > 20 A).
