  We report preliminary measurements of the exclusive charmless semileptonic
branching fractions of the B+ --> eta l+ nu and B+ --> eta' l+ nu decays. These
measurements are based on 316 fb-1 of data collected at the Y(4S) resonance by
the BABAR detector. In events in which the decay of one B meson to a hadronic
final state is fully reconstructed, the semileptonic decay of the recoiling B
meson is identified by the detection of a charged lepton and an eta or eta'. We
measure the branching fraction BR(B+ --> eta l+ nu) = (0.84 +/- 0.27 +/- 0.21)
x 10-4, where the first error is statistical and the second one systematic. We
also set an upper limit on the branching fraction of BR(B+ --> eta l+ nu) < 1.4
x 10-4 and BR(B+ --> eta' l+ nu) < 1.3 x 10^-4 at the 90% confidence level.
