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Global routing (GR) is a fundamental task in modern chip design and various learning techniques have been devised. However, a persistent challenge is the inherent lack of a mechanism to guarantee the routing connectivity in network's prediction results, necessitating post-processing search or reinforcement learning (RL) to enforce the connectivity. In this paper, we propose a neural GR solver called DSBRouter, leveraging the Diffusion Schrödinger Bridge (DSB) model for GR. During training, unlike previous works that learn the mapping from noise to routes, we establish a bridge between the initial pins and the routing via DSB, which learns the forward and backward mapping between them. For inference, based on the evaluation metric (e.g. low overflow), we further introduce a sampling scheme with evaluation-based guidance to enhance the routing predictions. Note that DSBRouter is an end-to-end model that does not require a post-step to ensure connectivity. Empirical results show that it achieves SOTA performance on the overflow reduction in ISPD98 and part of ISPD07. In some cases, DSBRouter can even generate routes with zero overflow.