The alignment of single-particle images fails at low signal-to-noise ratios and small particle sizes, because noise produces false peaks in the cross-correlation function used for alignment. A maximum-likelihood approach to the two-dimensional alignment problem is described which allows the underlying structure to be estimated from large data sets of very noisy images. Instead of finding the optimum alignment for each image, the algorithm forms a weighted sum over all possible in-plane rotations and translations of the image. The weighting factors, which are the probabilities of the image transformations, are computed as the exponential of a cross-correlation function. Simulated data sets were constructed and processed by the algorithm. The results demonstrate a greatly reduced sensitivity to the choice of a starting reference, and the ability to recover structures from large data sets having very low signal-to-noise ratios.