This paper proposes DropSignSGD, a communication-efficient and network-fault tolerant algorithm for training deep neural networks in a distributed and synchronous fashion. In DropSignSGD, all numerical elements communicated between machines are either 1 or -1, represented by only one bit. More importantly, DropSignSGD does not decline the benchmark accuracy on the ImageNet dataset when compared with the traditional distributed stochastic gradient descent algorithm, owing to a little trick in memorizing unused gradients. Experimental results are supported by a mathematical proof showing that DropSignSGD converges under standard assumptions.