A new reaction chamber has been designed and constructed to measure g-factors of short-lived excited states using the Transient Field technique in combination with Coulomb excitation in inverse kinematics. In this paper we will discuss several important aspects which have to be considered in order to successfully carry out this type of measurement with radioactive ion beams, instead of the stable beams used in a wide range of experiments in the past. The technical solutions to the problems arising from the use of such radioactive beams will be exposed in detail and the first successful experiment using the new chamber in combination with MINIBALL cluster detectors at REX-ISOLDE (CERN) will be reported on. &