Approximating model predictive control (MPC) using imitation learning (IL) allows for fast control without solving expensive optimization problems online. However, methods that use neural networks in a simple L2-regression setup fail to approximate multi-modal (set-valued) solution distributions caused by local optima found by the numerical solver or non-convex constraints, such as obstacles, significantly limiting the applicability of approximate MPC in practice. We solve this issue by using diffusion models to accurately represent the complete solution distribution (i.e., all modes) up to kilohertz sampling rates. This work shows that diffusion-based AMPC significantly outperforms L2-regression-based approximate MPC for multi-modal action distributions. In contrast to most earlier work on IL, we also focus on running the diffusion-based controller at a higher rate and in joint space instead of end-effector space. Additionally, we propose the use of gradient guidance during the denoising process to consistently pick the same mode in closed loop to prevent switching between solutions. We propose using the cost and constraint satisfaction of the original MPC problem during parallel sampling of solutions from the diffusion model to pick a better mode online. We evaluate our method on the fast and accurate control of a 7-DoF robot manipulator both in simulation and on hardware deployed at 250 Hz, achieving a speedup of more than 70 times compared to solving the MPC problem online and also outperforming the numerical optimization (used for training) in success ratio.