We have developed a new positron lifetime spectrometer. The spectrometer consists of a fast digital oscilloscope, BaF(2) scintillators and photomultipliers. The BaF(2) scintillators are shielded by conical lead collimators from Compton scattered gamma-rays which deteriorate the lifetime spectrum. The performance of the lead collimators is evaluated with EGS4. The lifetime of delocalized parapositronium in alpha-SiO(2) is successfully determined with the spectrometer.