Recent work on acoustic parameter estimation indicates that geometric room volume can be useful for modeling the character of an acoustic environment. However, estimating volume from audio signals remains a challenging problem. Here we propose using a convolutional neural network model to estimate the room volume blindly from reverberant single-channel speech signals in the presence of noise. The model is shown to produce estimates within approximately a factor of two to the true value, for rooms ranging in size from small offices to large concert halls.