During the development of the spinal cord, proliferative neural progenitors differentiate into postmitotic neurons with distinct fates. How cells switch from progenitor states to differentiated fates is poorly understood. To address this question, we studied the differentiation of progenitors in the zebrafish spinal cord, focusing on the differentiation of Kolmer-Agduhr3 (KA3) interneurons from lateral floor plate (LFP) progenitors. In vivo cell tracking demonstrates that KA3 cells are generated from LFP progenitors by both symmetric and asymmetric cell divisions. A photoconvertible reporter of signaling history (PHRESH) reveals distinct temporal profiles of Hh response: LFP progenitors continuously respond to Hh, while KA3 cells lose Hh response upon differentiation. Hh signaling is required in LFP progenitors for KA3 fate specification, but prolonged Hh signaling interferes with KA3 differentiation. Notch signaling acts permissively to maintain LFP progenitor cells: activation of Notch signaling prevents differentiation, whereas inhibition of Notch signaling results in differentiation of ectopic KA3 cells. These results indicate that neural progenitors depend on Notch signaling to maintain Hh responsiveness and rely on Hh signaling to induce fate identity, whereas proper differentiation depends on the attenuation of both Notch and Hh signaling.