The virtual machine (VM) scheduling problem in cloud brokers that support cloud bursting is fraught with uncertainty due to the on-demand nature of Infrastructure as a Service (IaaS) VMs. Until a VM request is received, the scheduler does not know in advance when it will arrive or what configurations it demands. Even when a VM request is received, the scheduler does not know when the VM's lifecycle expires. Existing studies begin to use deep reinforcement learning (DRL) to solve such scheduling problems. However, they do not address how to guarantee the QoS of user requests. In this paper, we investigate a cost optimization problem for online VM scheduling in cloud brokers for cloud bursting to minimize the cost spent on public clouds while satisfying specified QoS restrictions. We propose DeepBS, a DRL-based online VM scheduler in a cloud broker which learns from experience to adaptively improve scheduling strategies in environments with non-smooth and uncertain user requests. We evaluate the performance of DeepBS under two request arrival patterns which are respectively based on Google and Alibaba cluster traces, and the experiments show that DeepBS has a significant advantage over other benchmark algorithms in terms of cost optimization.