The Phase-Imaging Ion-Cyclotron-Resonance (PI-ICR) technique has been commissioned at the JYFLTRAP double Penning trap mass spectrometer. This technique is based on projecting the ion motion in the Penning trap onto a position-sensitive multichannel-plate ion detector. Mass measurements of stable 85 Rb $ ^{+}$ and 87 Rb $ ^{+}$ ions with well-known mass values show that relative uncertainties $ \Delta m/m \leq 7\cdot 10^{-10}$ are possible to reach with the PI-ICR technique at JYFLTRAP. The significant improvement both in resolving power and in precision compared to the conventional Time-of-Flight Ion Cyclotron Resonance technique will enable measurements of close-lying isomeric states and of more exotic isotopes as well as ultra-high precision measurements required, e.g., for neutrino physics. In addition, a new phase-dependent cleaning method based on the differences in the accumulated cyclotron motion phases has been demonstrated with short-lived 127 In $ ^{+}$ and 127m In $ ^{+}$ ions.