The concept of weak measurement was introduced by Aharonov, Albert and Vaidman in 1988, as a part of time-symmetric formulation of quantum mechanics. Assuming the coupling of the measured observable and the pointer is very weak, they showed that the shift of the pointer can be amplified by postselection performed on the data. This amplification has led to the measurements of very small effects using this protocol, as well as studies of a more foundational nature. However, weak measurement has so far been demonstrated using classical Gaussian laser beams. We demonstrate a new type of weak measurement based on the dynamics of spontaneous emission of an atom---a manifestly quantum phenomenon. The pointer in our scheme is given by the Lorentzian distribution characterizing atomic exponential decay via emission of a single photon, and the amplification is manifest as a large shift the mean arrival time of the emitted photon. The dynamics of spontaneous emission offer a broader view of the measurement process than is usually considered within the weak measurement formalism. Our scheme opens the path for the use of atoms and atomlike systems as sensitive probes in weak measurements.