Near-field heat transfer across a gap between plane-parallel tungsten layers in vacuo was studied experimentally with the temperature of the cold sample near 5 K and the temperature of the hot sample in the range 10-40 K as a function of the gap size d. At gaps smaller than one-third of the peak wavelength λ(m) given by Wien's displacement law, the near-field effect was observed. In comparison with blackbody radiation, hundred times higher values of heat flux were achieved at d≈1 μm. Heat flux normalized to the radiative power transferred between black surfaces showed scaling (λ(m)/d)(n), where n≈2.6. This Letter describes the results of experiment and a comparison with present theory over 4 orders of magnitude of heat flux.