Before revision:  The use of a discrete distribution is necessary in practice, as finite computers cannot  represent samples from continuous distributions and numerical errors may destroy the privacy guarantee .
After revision:  The use of a discrete distribution is necessary in practice, as finite computers cannot represent samples from continuous distributions , and previous work has demonstrated that seemingly innocuous numerical errors can entirely destroy privacy. Moreover, when the underlying data is itself discrete (e.g., population counts), adding continuous noise makes the result less interpretable. With these shortcomings in mind, we introduce and analyze the discrete Gaussian in the context of differential privacy. Specifically, we theoretically and experimentally show that adding discrete Gaussian noise provides essentially the same privacy and accuracy guarantees as the addition of continuous Gaussian noise. We also present an simple and efficient algorithm for exact sampling from this distribution. This demonstrates its applicability for privately answering counting queries, or more generally, low-sensitivity integer-valued queries .