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 ( T _ { eff } = 3950 \pm 150 K , \log g = 6.98 \pm 0.15 , R _ { WD } = ( 1.9 \pm 0.2 ) \times 10 ^ { 9 } cm ) fits our spectral data remarkably well for the known mass and distance ( M = 0.25 \pm 0.02 M _ { \odot } , d = 156.3 \pm 1.3 pc ) , yielding the white dwarf age ( \tau _ { WD } = 6.0 \pm 0.5 Gyr ) . In the UV , we find a spectral shape consistent with thermal emission from the bulk of the neutron star surface , with surface temperature between 1.25 \times 10 ^ { 5 } and 3.5 \times 10 ^ { 5 } K. The temprature 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 = 1.56 \pm 0.01 , E _ { cut } = 1.1 \pm 0.2 GeV ) that is seen in gamma-rays and perhaps even extends to the near-UV .