Network slicing is a promising technology that allows mobile network operators to efficiently serve various emerging use cases in 5G. It is challenging to optimize the utilization of network infrastructures while guaranteeing the performance of network slices according to service level agreements (SLAs). To solve this problem, we propose SafeSlicing that introduces a new constraint-aware deep reinforcement learning (CaDRL) algorithm to learn the optimal resource orchestration policy within two steps, i.e., offline training in a simulated environment and online learning with the real network system. On optimizing the resource orchestration, we incorporate the constraints on the statistical performance of slices in the reward function using Lagrangian multipliers, and solve the Lagrangian relaxed problem via a policy network. To satisfy the constraints on the system capacity, we design a constraint network to map the latent actions generated from the policy network to the orchestration actions such that the total resources allocated to network slices do not exceed the system capacity. We prototype SafeSlicing on an end-to-end testbed developed by using OpenAirInterface LTE, OpenDayLight-based SDN, and CUDA GPU computing platform. The experimental results show that SafeSlicing reduces more than 20% resource usage while meeting SLAs of network slices as compared with other solutions.