  We report on Keck and {\em Hubble Space Telescope} observations of the binary
millisecond pulsar PSR B1855+09. We detect its white-dwarf companion and
measure $\mv=25.90\pm0.12$ and $\mi=24.19\pm0.11$ (Vega system). From the
reddening-corrected color, $(\mv-\mi)_0=1.06\pm0.21$, we infer a temperature
$\Teff=4800\pm800 $K. The white-dwarf mass is known accurately from
measurements of the Shapiro delay of the pulsar signal,
$\Mcomp=0.258^{+0.028}_{-0.016} \Msun$. Hence, 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 is set by the thickness of the hydrogen layer
surrounding the helium core. From the properties of similar systems, it has
been inferred that helium white dwarfs form with thick hydrogen layers, with
mass $\simgt3\times10^{-3} \Msun$, which leads to significant additional
heating. This is consistent with expectations from simple evolutionary models
of the preceding binary evolution. For PSR B1855+09, though, such models lead
to a cooling age of $\sim10 $Gyr, which is twice the spin-down age of the
pulsar. It could be that the spin-down age were incorrect, which would call the
standard vacuum dipole braking model into question. For two other pulsar
companions, however, ages well over 10 Gyr are inferred, indicating that the
problem may lie with the cooling models. There is no age discrepancy for models
in which the white dwarfs are formed with thinner hydrogen layers
($\simlt3\times10^{-4} \Msun$).
