A reaction microscope (ReMi) has been combined with a magneto-optical trap (MOT) for the kinematically complete investigation of atomic break-up processes. With the novel MOTReMi apparatus, the momentum vectors of the fragments of laser-cooled and state-prepared lithium atoms are measured in coincidence and over the full solid angle. The first successful implementation of a MOTReMi could be realized due to an optimized design of the present setup, a nonstandard operation of the MOT, and by employing a switching cycle with alternating measuring and trapping periods. The very low target temperature in the MOT (~2 mK) allows for an excellent momentum resolution. Optical preparation of the target atoms in the excited Li 22P3/2 state was demonstrated providing an atomic polarization of close to 100%. While first experimental results were reported earlier, in this work, we focus on the technical description of the setup and its performance in commissioning experiments involving target ionization in 266 nm laser pulses and in collisions with projectile ions.