  The aim of this paper is to develop a sequence of discrete approximations to
a one-dimensional It\^o diffusion that almost surely converges to a weak
solution of the given stochastic differential equation. Under suitable
conditions, the solution of the stochastic differential equation can be reduced
to the solution of an ordinary differential equation plus an application of
Girsanov's theorem to adjust the drift. The discrete approximation is based on
a specific strong approximation of Brownian motion by simple, symmetric random
walks (the so-called "twist and shrink" method). A discrete It\^o's formula is
also used during the discrete approximation.
