A topic of great current interest is the capacity of populations to adapt genetically to rapidly changing climates, for example by evolving the timing of life-history events, but this is challenging to address experimentally. I use a plant invasion as a model system to tackle this question by combining molecular markers, a common garden experiment and climatic niche modelling. This approach reveals that non-native Lactuca serriola originates primarily from Europe, a climatic subset of its native range, with low rates of admixture from Asia. It has rapidly refilled its climatic niche in the new range, associated with the evolution of flowering phenology to produce clines along climate gradients that mirror those across the native range. Consequently, some non-native plants have evolved development times and grow under climates more extreme than those found in Europe, but not among populations from the native range as a whole. This suggests that many plant populations can adapt rapidly to changed climatic conditions that are already within the climatic niche space occupied by the species elsewhere in its range, but that evolution to conditions outside of this range is more difficult. These findings can also help to explain the prevalence of niche conservatism among non-native species.