  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. %Earlier attempts to realize this
combination failed due to intrinsic incompatibilities of the magnetic fields
required for MOT and ReMi. 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 ($2mK$) allow for an excellent momentum resolution. Optical preparation
of the target atoms in the excited Li $2^2P_{3/2}$ state was demonstrated
providing an atomic polarization of close to 100\percent. 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 $266nm$ laser pulses and in
collisions with projectile ions.
