We specify a stochastic economy-climate model, adapting Nordhaus' deterministic economy-climate model by allowing for Weitzman-type stochasticity. We show that, under expected power utility, the model is fragile to heavy-tailed distributional assumptions and we derive necessary and sufficient conditions on the utility function to avoid fragility. We solve our stochastic economy-climate model for two cases with compatible pairs of utility functions and heavy-tailed distributional assumptions. We further develop and implement a procedure to learn the input parameters of our model and show that the model thus specified produces robust optimal policies. The numerical results indicate that higher levels of uncertainty lead to less abatement and consumption, and to more investment.