A new technique for range verification in proton beam therapy has been developed. It is based on the detection of the prompt γ rays that are emitted naturally during the delivery of the treatment. A spectrometer comprising 16 LaBr3(Ce) detectors in a symmetrical configuration is employed to record the prompt γ rays emitted along the proton path. An algorithm has been developed that takes as inputs the LaBr3(Ce) detector signals and reconstructs the maximum γ-ray intensity peak position, in full 3 dimensions. For a spectrometer radius of 8 cm, which could accommodate a paediatric head and neck case, the prompt γ-ray origin can be determined from the width of the detected peak with a σ of 4.17 mm for a 180 MeV proton beam impinging a water phantom. For spectrometer radii of 15 and 25 cm to accommodate larger volumes this value increases to 5.65 and 6.36 mm. For a 8 cm radius, with a 5 and 10 mm undershoot, the σ is 4.31 and 5.47 mm. These uncertainties are comparable to the range uncertainties incorporated in treatment planning. This work represents the first step towards a new accurate, real-time, 3D range verification device for spot-scanning proton beam therapy.