  We present extensive spectral and photometric observations of the recycled
pulsar/white-dwarf binary containing PSR J0437-4715, which we analyzed together
with archival X-ray and gamma-ray data, to obtain the complete mid-infrared to
gamma-ray spectrum. We first fit each part of the spectrum separately, and then
the whole multi-wavelength spectrum. We find that the optical-infrared part of
the spectrum is well fit by a cool white dwarf atmosphere model with pure
hydrogen composition. The model atmosphere (Teff = 3950pm150K, log
g=6.98pm0.15, R_WD=(1.9pm0.2)e9 cm) fits our spectral data remarkably well for
the known mass and distance (M=0.25pm0.02Msun, d=156.3pm1.3pc), yielding the
white dwarf age (tau=6.0pm0.5Gyr). In the UV, we find a spectral shape
consistent with thermal emission from the bulk of the neutron star surface,
with surface temperature between 1.25e5 and 3.5e5K. The temperature of the
thermal spectrum suggests that some heating mechanism operates throughout the
life of the neutron star. The temperature distribution on the neutron star
surface is non-uniform. In the X-rays, we confirm the presence of a high-energy
tail which is consistent with a continuation of the cut-off power-law component
(Gamma=1.56pm0.01, Ecut=1.1pm0.2GeV) that is seen in gamma-rays and perhaps
even extends to the near-UV.
