We report experimental results for the thermal boundary resistance RK between gold and superfluid 4He which were obtained by using thermometry with three-nanokelvin resolution. The data imply that RK is singular at the superfluid transition temperature Tॕ. For t蠁1-T/Tॕ\lesssim10-4, RK becomes noticeably dependent on the heat flux Q used to make the measurement. The reduced temperature tc at which this nonlinear resistance is first observed is approximately proportional to Q. For t蠄tc dRK/dQ is approximately proportional to 1/t.