  We present a study of the galaxy population in the cluster RX J1053.7+5735,
one of the most distant X-ray selected clusters of galaxies, which also shows
an unusual double-lobed X-ray morphology, indicative of a possible equal-mass
cluster merger. Using Keck-DEIMOS spectroscopic observations of galaxies in the
2x1.5 arcmin region surrounding RX J1053.7+5735, we secured redshifts for six
galaxies in the range 1.129 < z < 1.139, with a mean redshift <z>=1.134. The
mean redshift agrees well with the cluster X-ray redshift previously estimated
from the cluster X-ray Fe-K line, confirming the presence of a cluster at
z~1.135. Galaxies with concordant redshifts are located in both eastern and
western sub-clusters of the double cluster structure, indicating that both
sub-clusters are at similar redshifts. This result is also consistent with a
previous claim that both eastern and western X-ray lobes have similar X-ray
redshifts. Based on their separation of ~ 250 kpc/h, these results support the
interpretation that RX J1053.7+5735 is an equal-mass cluster merger taken place
at z ~ 1, although further direct evidence for dynamical state of the cluster
is needed for a more definitive statement about the cluster merging state. The
six galaxies have a line-of-sight velocity dispersion = 650 km/s. All six
galaxies show clear absorption features of CaII H & K, and several Balmer
lines, typical of early galaxies at the present epoch, in agreement with their
I-K colors. A color-magnitude diagram, constructed from deep optical/NIR
observations of the RX J1053.7+5735 field, shows a clear red color sequence.
There is an indication that the red sequence in RX J1053.7+5735 lies ~ 0.3 to
the blue of the Coma line, qualitatively consistent with previous studies
investigating other clusters at z ~ 1.
