We study the Gaussian multiple channel with random user activity, in the regime where the number of users is proportional to the code length. The receiver may know some statistics about the number of active users, but does not know the exact number nor the identities of the active users. We first derive achievability bounds on the error probabilities by analyzing Gaussian random codebooks with maximum-likelihood decoding. We then propose an efficient CDMA-type scheme based on a spatially coupled signature matrix and approximate message passing (AMP) decoding. Rigorous asymptotic guarantees on the error performance of the AMP decoder are derived. A numerical comparison indicates that the asymptotic error guarantees of the spatially coupled scheme are significantly better than those obtained via the finite-length achievability bounds.