  We study the thermalization and the Bose-Einstein condensation of a paraxial,
spectrally narrow beam of quantum light propagating in a lossless bulk Kerr
medium. The spatiotemporal evolution of the quantum optical field is ruled by a
Heisenberg equation analogous to the quantum nonlinear Schr\"odinger equation
of dilute atomic Bose gases. Correspondingly, in the weak-nonlinearity regime,
the phase-space density evolves according to the Boltzmann equation.
Expressions for the thermalization time and for the temperature and the
chemical potential of the eventual Bose-Einstein distribution are found. After
discussing experimental issues, we introduce an optical setup allowing the
evaporative cooling of a guided beam of light towards Bose-Einstein
condensation. This might serve as a novel source of coherent light.
