We examine if extreme weather exposure impacts firms' cost of equity. Motivated by a consumption-based asset pricing model with heterogeneous agents, we reveal the existence of an extreme weather risk premium in the cross-section of stock returns. In the period from 1995 to 2019, domestic U.S. stocks with the most negative sensitivity to thunderstorm losses earned excess returns of 6.5% p.a. over those with the most positive sensitivity. This premium can neither be explained by risk factors from standard asset pricing models nor by firm characteristics. Our results reveal a novel link between climate risk and firm value.