We present a square root limit on low probability of detection (LPD) communication over additive white Gaussian noise (AWGN) channels. Specifically, if a warden has an AWGN channel to the transmitter with non-zero noise power, we prove that o(√n) bits can be sent from the transmitter to the receiver in n AWGN channel uses with probability of detection by the warden less than e for any ∊ > 0, and, if a lower bound on the noise power on the warden's channel is known, then O(√n) bits can be covertly sent in n channel uses. Conversely, trying to transmit more than O(√n) bits either results in detection by the warden with probability one or a non-zero probability of decoding error as n → ∞. Further, we show that LPD communication on the AWGN channel allows one to send a nonzero symbol on every channel use, in contrast to what might be expected from the square root law found recently in image-based steganography.