  We have identified a merging galaxy cluster with evidence of two distinct
sub-clusters. The X-ray and optical data suggest that the subclusters are
moving away from each other after closest approach. This cluster merger was
discovered from observations of the well localized short-duration gamma-ray
burst (GRB), GRB 050509B. The Swift/Burst Alert Telescope (BAT) source position
is coincident with a cluster of galaxies ZwCl 1234.0+02916. The subsequent
Swift/X-Ray Telescope (XRT) localization of the X-ray afterglow found the GRB
coincident with 2MASX J12361286+2858580, a giant red elliptical galaxy in the
cluster. Deep multi-epoch optical images were obtained to constrain the
evolution of the GRB afterglow, including a 27480s exposure in the F814W band
with Hubble Space Telescope Advanced Camera for Surveys (ACS), among the
deepest imaging ever obtained towards a known galaxy cluster in a single
passband. We perform a weak gravitational lensing analysis, including mapping
the total mass distribution of the merger system. Combined with Chandra X-ray
Observatory and Swift/XRT observations, we investigate the dynamical state of
the merger to probe the nature of the dark matter component. Our weak
gravitational lensing measurements reveal a separation of the X-ray centroid of
the western subcluster from the center of the mass and galaxy light
distributions, similar to that of the famous "Bullet cluster". We conclude that
the "Burst cluster" is another candidate merger system for determining the
nature of dark matter and for studying the environment of short GRBs. We
discuss connections between the cluster dynamical state and/or matter
composition and compact object mergers, the leading model for the origin of
short GRBs. Finally, we present results from a weak lensing survey based on
archival Very Large Telescope (VLT) images in the areas of 5 other short GRBs.
