  We report on the adiabatic splitting of a BEC of $^{87}$Rb atoms by an
asymmetric double-well potential located above the edge of a perpendicularly
magnetized TbGdFeCo film atom chip. By controlling the barrier height and
double-well asymmetry the sensitivity of the axial splitting process is
investigated through observation of the fractional atom distribution between
the left and right wells. This process constitutes a novel sensor for which we
infer a single shot sensitivity to gravity fields of $\delta
g/g\approx2\times10^{-4}$. From a simple analytic model we propose improvements
to chip-based gravity detectors using this demonstrated methodology.
