  We propose a new numerical method for one dimensional stochastic differential
equations (SDEs). The main idea of this method is based on a representation of
a weak solution of a SDE with a time changed Brownian motion, dated back to
Doeblin (1940). In cases where the diffusion coefficient is bounded and
$\beta$-H\"{o}lder continuous with $0 < \beta \leq 1$, we provide the rate of
strong convergence. An advantage of our approach is that we approximate the
weak solution, which enables us to treat a SDE with no strong solution. Our
scheme is the first to achieve the strong convergence for the case $0 < \beta <
1/2$.
