We study the convergence of a drift implicit scheme for one-dimensional SDEs
that was considered by Alfonsi for the Cox-Ingersoll-Ross (CIR) process. Under
general conditions, we obtain a strong convergence of order 1. In the CIR case,
Dereich, Neuenkirch and Szpruch have shown recently a strong convergence of
order 1/2 for this scheme. Here, we obtain a strong convergence of order 1
under more restrictive assumptions on the CIR parameters
