We report high-resolution measurements of the singular contributionRbto the thermal boundary resistance between a solid surface and superfluid helium near the superfluid-transition temperatureT&lgr;.The results confirm the observation by Murphy and Meyer that a gap between the cell end and the sidewall leads to an apparent finite-current contribution toRb.In the absence of such a gap, overall agreement ofRbwith theoretical predictions is very good. Remaining small differences require further investigations. Without a sidewall gap and within our resolution we found no finite-current effects over the range3.9 &mgr;W/cm2<Q<221 &mgr;W/cm2.©1998 American Institute of Physics.