Plasma wakefield accelerators are capable of sustaining gigavolt-per-centimeter accelerating fields, surpassing the electric breakdown threshold in state-of-the-art accelerator modules by 3-4 orders of magnitude. Beam-driven wakefields offer particularly attractive conditions for the generation and acceleration of high-quality beams. However, this scheme relies on kilometer-scale accelerators. Here, we report on the demonstration of a millimeter-scale plasma accelerator powered by laser-accelerated electron beams. We showcase the acceleration of electron beams to 128 MeV, consistent with simulations exhibiting accelerating gradients exceeding 100 GV m-1. This miniaturized accelerator is further explored by employing a controlled pair of drive and witness electron bunches, where a fraction of the driver energy is transferred to the accelerated witness through the plasma. Such a hybrid approach allows fundamental studies of beam-driven plasma accelerator concepts at widely accessible high-power laser facilities. It is anticipated to provide compact sources of energetic high-brightness electron beams for quality-demanding applications such as free-electron lasers.