  The traditional view that Ly-alpha emission and dust should be mutually
exclusive has been questioned more and more often; most notably, the
observations of Ly-alpha emission from ULIRGs seem to counter this view. In
this paper we seek to address the reverse question. How large a fraction of
Ly-alpha selected galaxies are ULIRGs? Using two samples of 24/25 Ly-alpha
emitting galaxies at z = 0.3/2.3, we perform this test, including results at z
= 3.1, and find that, whereas the ULIRG fraction at z = 3.1 is very small, it
systematically increases towards lower redshifts. There is a hint that this
evolution may be quite sudden and that it happens around a redshift of z ~ 2.5.
After measuring the infrared luminosities of the Ly-alpha emitters, we find
that they are in the normal to ULIRG range in the lower redshift sample, while
the higher redshift galaxies all have luminosities in the ULIRG category. The
Ly-alpha escape fractions for these infrared bright galaxies are in the range
1-100 % in the z = 0.3 galaxies, but are very low in the z = 2.3 galaxies, 0.4
% on average. The unobscured star formation rates are very high, ranging from
500 to more than 5000 M_sun/yr, and the dust attenuation derived are in the
range 0.0 < A_V < 3.5.
