  We determine the energy splitting of the conduction-band valleys in
two-dimensional electrons confined to low-disorder Si quantum wells. We probe
the valley splitting dependence on both perpendicular magnetic field $B$ and
Hall density by performing activation energy measurements in the quantum Hall
regime over a large range of filling factors. The mobility gap of the
valley-split levels increases linearly with $B$ and is strikingly independent
of Hall density. The data are consistent with a transport model in which valley
splitting depends on the incremental changes in density $eB/h$ across quantum
Hall edge strips, rather than the bulk density. Based on these results, we
estimate that the valley splitting increases with density at a rate of 116
$\mu$eV/10$^{11}$cm$^{-2}$, consistent with theoretical predictions for
near-perfect quantum well top interfaces.
