96Zr and 48Ca are unique among double-β decay candidate nuclides in that they may also undergo single-β decay. In the case of 96Zr, the single-β decay mode is dominated by the fourth-forbidden β decay with a 119 keV Q value. A search was conducted for the β decay of 96Zr by observing the decay of the daughter 96Nb nucleus. Two coaxial high-purity germanium detectors were used in coincidence to detect the γ-ray cascade produced by the daughter nucleus as it de-excited to the ground state. The experiment was carried out at the Kimballton Underground Research Facility and produced 685.7 days of data with a 17.91 g enriched sample. No counts were seen above background, producing a limit of T1/2 > 2.4 ×1019 year. This is the first experimental search that is able to discern between the β decay and the double-β decay to an excited state of 96Zr.