Using drift-scan data, a new approach to determining surface brightness pro
files, and techniques for detecting low surface brightness signals, we fit
the light profile of the brightest cluster galaxy (BCG) in the rich cluster
Abell 1651 out to 670 h(-1) kpc. This radius is a significant fraction of
the virial radius of the cluster (2 h(-1) Mpc), indicating that the sizes o
f the BCG and the cluster are comparable. We find that the profile is consi
stent with a de Vaucouleurs profile over the radial range probed. We also f
ind that the integrated light profile of the BCG in Abell 1651 contributes
36% of the total cluster light within 500 h(-1) kpc. Including all luminous
components, we obtain M/L-I similar to 160 h for the cluster, which would
be overestimated by similar to 20% without the BCG halo. Furthermore, the r
elatively red color of the BCG at large radii suggests that recent disrupti
on and tidal stripping of spirals and dwarf ellipticals do not contribute s
ignificantly to the halo luminosity. The color and the form of the profile
are consistent with a scenario in which the BCG forms from filamentary coll
apse during the epoch of cluster formation, with relatively little evolutio
n in the past 5 Gyr. We remove the BCG and other detected galaxies from the
image and construct a two-dimensional surface brightness map of the cluste
r core. Several knots of excess emission are found, but the total diffuse c
omponent is constrained to contribute less than 5% of the cluster light.