Allowing humans and robots to interact in close proximity to each other has great potential for increasing the effectiveness of human-robot teams across a large variety of domains. However, as we move toward enabling humans and robots to interact at ever-decreasing distances of separation, effective safety technologies must also be developed. While new, inherently human-safe robot designs have been established, millions of industrial robots are already deployed worldwide, which makes it attractive to develop technologies that can turn these standard industrial robots into human-safe platforms. In this work, we present a real-time safety system capable of allowing safe human-robot interaction at very low distances of separation, without the need for robot hardware modification or replacement. By leveraging known robot joint angle values and accurate measurements of human positioning in the workspace, we can achieve precise robot speed adjustment by utilizing real-time measurements of separation distance. This, in turn, allows for collision prevention in a manner comfortable for the human user. We demonstrate our system achieves latencies below 9.64 ms with 95% probability, 11.10 ms with 99% probability, and 14.08 ms with 99.99% probability, resulting in robust real-time performance.