Secondary companions of rapidly spinning pulsars in low-mass systems are expected to be white dwarfs. Observations of these white dwarfs are expected to provide critical clues to the evolution of magnetic fields of neutron stars and stellar evolution scenarios as well. We report new radio and optical observations of the 5.4 ms binary pulsar 1855 + 09. Using H I and CO observations we establish the visual extinction to the system to be about 1.5 mag. Recent timing observations by Ryba et al. have established that the companion or the secondary star has a mass of 0.22 M. and that the system is located more than 290 pc. We find only one star brighter than R approximately 24.6 that, on the basis of positional coincidence, can be plausibly the optical counterpart of the secondary star. The spectrum of this candidate is inconsistent with a low-mass main-sequence star. Neither is it a white dwarf because the spectroscopic distance modulus is inconsistent with the lower limit on distance obtained from timing observations. The true optical counterpart must be fainter than our detection limit: R > 24.6 and I > 23.4. Thus we conclude, as did Callanan et al., that the companion of PSr 1855 + 09 must be a low-mass cold white dwarf. At the nominal dispersion measure distance of 400 pc, all theoretical models predict a detectable white dwarf, inconsistent with our observations. However, at the larger distance of 800 pc, favored by the H I data, the upper limits are consistent with an old white dwarf. The inferred cooling age, while model-dependent, is comparable to 5 x 10(9) yr, the characteristic age of the pulsar. This result supports the hypothesis that magnetic field strengths of millisecond pulsars are essentially constant and that millisecond pulsars are long-lived objects. Further improvements in the parallax measurement and deeper images of the field are needed to understand the precise extent of te disagreement between different low-mass white dwarf cooling models and observations.