In this paper, we study the average controllability of a random heat equation, with the diffusivity serving as the random variable drawn from a general probability distribution. We show that the solutions of such random heat equations are both null and approximately controllable in average from an arbitrary open set of the domain and in an arbitrarily small time, recovering the known result when the random diffusivity is uniformly or exponentially distributed. (C) 2019 Elsevier Ltd. All rights reserved.