The behavior of the superfluid - normal fluid interface in hHe has been studied in the presence of a heat flux, Q→, in the range 0.1 < Q < 6 egr s−1 cm−2. Fourier’s Law, Q→ = κ▽ T, has been observed, for the first time, to break down very close to the interface. In the nonlinear region, κ depends on Q. Theoretical predictions, which do not take gravity into account, are compared with these experimental results. The predicted extent of the nonlinear region agrees well with the data; however, at lower Q, a systematic discrepancy is found that may be ascribed to the effect of gravity on the interface.