With low hardware cost and power consumption, direction-of-arrival (DOA) estimation exploiting one-bit quantized array data has become an attractive topic. In this paper, the gridless compressed sensing (CS) with multiple measurement vectors for one-bit DOA estimation is investigated. First, the one-bit quantized signal model is established. Then an atomic norm minimization scheme is proposed based on the output of one-bit quantizer, in which the objective function is reformulated as a semidefinite programming problem combined with a one-sided $$l_1$$ -norm constraint, representing the sign inconsistency between the quantized and unquantized measurements. To efficiently solve such a problem and reduce its computational complexity, an accelerated proximal gradient-based algorithm is developed. The proposed approach outperforms the one-bit MUSIC in a small number of measurements, and avoids the grid-mismatch issue of several existing one-bit CS methods. Numerical experiments are conducted to validate the superiorities of the proposed one-bit DOA estimation approach in accuracy and running time.