We describe the measurement process for a model system whose Schrödinger equation yields to explicit solution. Any device for an effectively irreversible measurement must not just be a ``meter'' with a single degree of freedom; it must also contain a ``heat bath'' with many degrees of freedom. Such a bath may cause an effectively instantaneous decay of quantum coherences between macroscopically distinct meter readings, i.e., impart practically classical behavior to the meter. In turn, such a meter can be read without thereby suffering back reactions. We establish the limits in which a meter reading is uniquely related, with reasonable certainty, to a definite eigenvalue of the measured observable.