Recently, score-based generative models have been successfully employed for the task of speech enhancement. A stochastic differential equation is used to model the iterative forward process, where at each step environmental noise and white Gaussian noise are added to the clean speech signal. While in limit the mean of the forward process ends at the noisy mixture, in practice it stops earlier and thus only at an approximation of the noisy mixture. This results in a discrepancy between the terminating distribution of the forward process and the prior used for solving the reverse process at inference. In this paper, we address this discrepancy and propose a forward process based on a Brownian bridge. We show that such a process leads to a reduction of the mismatch compared to previous diffusion processes. More importantly, we show that our approach improves in objective metrics over the baseline process with only half of the iteration steps and having one hyperparameter less to tune.