  We investigate the phase structure of the random-field Ising model with a
bimodal random field distribution. Our aim is to test for the possibility of an
equilibrium spin-glass phase, and for replica symmetry breaking (RSB) within
such a phase. We study a low-temperature region where the spin-glass phase is
thought to occur, but which has received little numerical study to date. We use
the exchange Monte-Carlo technique to acquire equilibrium information about the
model, in particular the $P(q)$ distribution and the spectrum of eigenvalues of
the spin-spin correlation matrix (which tests for the presence of RSB). Our
studies span the range in parameter space from the ferromagnetic to the
paramagnetic phase. We find however no convincing evidence for any equilibrium
glass phase, with or without RSB, between these two phases. Instead we find
clear evidence (principally from the $P(q)$ distribution) that there are only
two phases at this low temperature, with a discontinuity in the magnetization
at the transition like that seen at other temperatures.
