We study the evolution of the probability density of an asexual, one locus population
under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation
with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws.
It is readily shown that no classical or standard weak solution definition yields solvability of the
problem. We provide an appropriate definition of weak solution for the problem, for which we show
existence and uniqueness. The solution displays a very distinctive structure and, for large time, we
show convergence to a unique stationary solution that turns out to be a singular measure supported
at the endpoints. An exponential rate of convergence to this steady state is also proved.