In this paper, we develop an online method that leverages machine learning to obtain feasible solutions to the AC optimal power flow (OPF) problem with negligible optimality gaps on extremely fast timescales (e.g., milliseconds), bypassing solving an AC OPF altogether. This is motivated by the fact that as the power grid experiences increasing amounts of renewable power generation, controllable loads, and other inverter-interfaced devices, faster system dynamics and quicker fluctuations in the power supply are likely to occur. Currently, grid operators typically solve AC OPF every 15 minutes to determine economic generator settings while ensuring grid constraints are satisfied. Due to the computational challenges with solving this nonconvex problem, many efforts have focused on linearizing or approximating the problem in order to solve the AC OPF on faster timescales. However, many of these approximations can be fairly poor representations of the actual system state and still require solving an optimization problem, which can be time consuming for large networks. In this work, we leverage historical data to learn a mapping between the system loading and optimal generation values, enabling us to find near-optimal and feasible AC OPF solutions on extremely fast timescales without actually solving an optimization problem.