Climate change is predicted to reduce crop productivity in several world regions. A growing literature has examined climate change impacts on crop yields by statistically estimating the historical relationship between weather variables and yield and projecting it into predicted future climate. We estimate a flexible statistical model using panel data from France over the period 1950-2014 to investigate the impacts of climate change on winter wheat, winter barley, and spring barley yields. For winter crops, our model captures the differential impacts of weather on yield growth over cold (fall-winter) and warm (spring-summer) seasons. Temperatures above 33ºC during the warm season appear harmful to all three crops. For winter crops, cold-season temperatures have a negligible effect on crop growth. Cereal yields are predicted to decline due to climate change under a wide range of climate models and emissions scenarios. Impacts are almost exclusively driven by increased heat exposure during the warm season. Under the most rapid warming scenario (RCP8.5) and holding growing areas constant, our model ensemble predicts a 16% decline in winter wheat yield, a 20% decline in winter barley yield, and a 42% decline in spring barley yield by the end of the century. Under this scenario, uncertainty stemming from climate model projections clearly dominates that stemming from the historically estimated climate-yield relationship. A comparison of our results with those from a recent study for Kansas wheat points to the critical role of local climatology on the marginal yield response to extreme temperature exposure.