Let γn be the standard Gaussian measure on R n. We prove that for every symmetric convex sets K, L in R n and every λ ∈ (0, 1), γn λK + (1 − λ)L 1 n λγn(K) 1 n + (1 − λ)γn(L) 1 n , thus settling a problem raised by Gardner and Zvavitch (2010). This is the Gaussian analogue of the classical Brunn-Minkowski inequality for the Lebesgue measure. We also show that, for a fixed λ ∈ (0, 1), equality is attained if and only if K = L.