We report on Keck and HST observations of the binary millisecond pulsar PSR B1855+09. We detect its white-dwarf companion and measure mF555W = 25.90 ± 0.12 and mF814W = 24.19 ± 0.11 (Vega system). From the reddening-corrected color we infer a temperature Teff = 4800 ± 800 K. The companion mass is known accurately from measurements of the Shapiro delay of the pulsar signal, . Given a cooling model, one can use the measured temperature to determine the cooling age. The main uncertainty in the cooling models for such low-mass white dwarfs is the amount of residual nuclear burning, which depends on the thickness of the hydrogen layer surrounding the helium core. For PSR B1855+09, such models lead to a cooling age of ∼10Gyr, which is twice the spin-down age of the pulsar. It may be that the pulsar does not brake (n=3.0) like a dipole rotating in vacuo. For other pulsar companions, however, ages well over lOGyr are inferred, indicating that the problem may lie with the cooling models. There is no age discrepancy for models in which the white dwarfs are formed with thinner hydrogen layers (< 3 × 10−4M⊙). See van Kerkwijk et al. ApJ (submitted) for more details.