The neutron lifetime is important in understanding the production of light nuclei in the first minutes after the big bang and it provides basic information on the charged weak current of the standard model of particle physics. Two different methods have been used to measure the neutron lifetime: disappearance measurements using bottled ultracold neutrons and decay rate measurements using neutron beams. The best measurements using these two techniques give results that differ by nearly 4 standard deviations. In this paper we describe a new method for measuring surviving neutrons in neutron lifetime measurements using bottled ultracold neutrons that provides better characterization of systematic uncertainties and enables higher precision than previous measurement techniques. We present results obtained using our method.