When does a drop, deposited on a cold substrate, stop spreading? Despite the practical relevance of this question, for example, in airplane icing and three-dimensional metal printing, the detailed mechanism of arrest in solidifying spreading drops has remained debated. Here we consider the spreading and arrest of hexadecane drops of constant volume on two smooth wettable substrates: copper with a high thermal conductivity and glass with a low thermal conductivity. We record the spreading radius and contact angle in time for a range of substrate temperatures. The experiments are complemented by a detailed analysis of the temperature field near the rapidly moving contact line, by means of similarity solutions of the thermohydrodynamic problem. Our combined experimental and theoretical results provide strong evidence that the spreading of solidifying drops is arrested when the liquid at the contact line reaches a critical temperature, which is determined by the effect of kinetic undercooling.