  In this note we propose an exact simulation algorithm for the solution of
dX_t=dW_t+b(X_t)dt (1) where b is a smooth real function except at point 0
where b(0+)\neq b(0-). The main idea is to sample an exact skeleton of X using
an algorithm deduced from the convergence of the solutions of the skew
perturbed equation dX^\beta_t=dW_t+b(X^\beta_t)dt + \beta dL^0_t {X^\beta} (2)
towards X solution of (1) as \beta tends to 0. In this note, we show that this
convergence induces the convergence of exact simulation algorithms proposed by
the authors in \cite{etoremartinez1} for the solutions of (2) towards a limit
algorithm. Thanks to stability properties of the rejection procedures involved
as \beta tends to 0, we prove that this limit algorithm is an exact simulation
algorithm for the solution of the limit equation (1). Numerical examples are
shown to illustrate the performance of this exact simulation algorithm.
