We obtained a light curve for the 5.75 ms pulsar J0437 -4715 in the 65-120 A range with 0.5 ms time resolution using the Deep Survey instrument on the Extreme-Ultraviolet Explorer (EUVE) satellite. The single-peaked profile has a pulsed fraction of 0.27 ± 0.05, similar to the ROSAT data in the overlapping energy band. A combined analysis of the EUVE and ROSAT data is consistent with a power-law spectrum of energy index α = 1.2-1.5, intervening column density N_H = (5-8) x 10~(19) cm~(-2), and luminosity 5.0 x 10~(30) ergs s~(-1) in the 0.1-2.4 keV band. We also use a bright EUVE/ROSAT source only 4.2′ from the pulsar, the Seyfert galaxy RX J0437.4-4711 ( = EUVE J0437-471 = 1ES 0435-472), to obtain an independent upper limit on the intervening absorption to the pulsar, N_H < 1.2 x 10~(20) cm~(-2). Although a blackbody spectrum fails to fit the ROSAT data, two-component spectral fits to the combined EUVE/ROSAT data are used to limit the temperatures and surface areas of thermal emission that might make partial contributions to the flux. A hot polar cap of radius 50-600 m and temperature (1.0-3.3) x 10~6 K could be present. Alternatively, a larger region with T = (4-12) x 10~5 K and area less than 200 km~2 might contribute most of the EUVE and soft X-ray flux, but only if a hotter component were present as well. Any of these temperatures would require some mechanism(s) of surface reheating to be operating in this old pulsar, the most plausible being the impact of accelerated electrons and positrons onto the polar caps. The kinematically corrected spin-down power of PSR J0437—4715 is only 4 x 10~(33) ergs s~(-1), which is an order of magnitude less than that of the lowest luminosity γ-ray pulsars Geminga and PSR B1055 ?52. The absence of high-energy γ-rays from PSR J0437—4715 might signify an inefficient or dead outer gap accelerator, which in turn accounts for the lack of a more luminous reheated surface such as those intermediate-age γ-ray pulsars may have.
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