A position-sensitive microchannel plate (MCP) detector has been installed at the Canadian Penning Trap (CPT) mass spectrometer located at the CAlifornium Rare Isotope Breeder Upgrade (CARIBU) facility at Argonne National Laboratory in order to carry out Phase-Imaging Ion Cyclotron Resonance (PI-ICR) measurements. With this new measurement method, proof-of-principle mass measurements of five nuclei were made to a precision of $\delta m/m \approx 10^{-7}$. The PI-ICR results are found to be consistent with previous Time-of-Flight Ion Cyclotron Resonance (ToF-ICR) measurements. The content of this thesis covers the entire mass measurement process beginning with beam production at CARIBU through to ion detection at the CPT and a comparison of the ToF-ICR and PI-ICR measurement methods. The future of mass measurements at the CPT with this new technique will also be discussed. / May 2016