PET imaging is a non-invasive technique for particle range verification in proton therapy. It is based on measuring the beta+ annihilations caused by nuclear interactions of the protons in the patient. In this work we present measurements for proton range verification in phantoms, performed at the CNAO particle therapy treatment center in Pavia, Italy, with our 10 x 10 cm^2 planar PET prototype DoPET. PMMA phantoms were irradiated with mono-energetic proton beams and clinical treatment plans, and PET data were acquired during and shortly after proton irradiation. We created 1-D profiles of the beta+ activity along the proton beam-axis, and evaluated the difference between the proximal rise and the distal fall-off position of the activity distribution. A good agreement with FLUKA Monte Carlo predictions was obtained. We also assessed the system response when the PMMA phantom contained an air cavity. The system was able to detect these cavities quickly after irradiation.