The capacity region of the two-user additive white Gaussian noise (AWGN) multiple-access channel (MAC) with user cooperation is studied. This channel differs from the classical AWGN MAC in that here each transmitter observes a noisy version of the channel inputs sent by the other transmitter. A new achievable region is presented based on a coding scheme where each transmitter sends a linear combination of these observations and of the codeword it produced to encode its message. For certain choices of the parameters our scheme can be viewed as a scheme where the messages are encoded into stationary processes and where the encoders apply linear time-invariant filters to their observations. The rates achieved by this latter scheme can be expressed in terms of the power spectra of the filters and input processes. For most choices of filters, the optimal input spectra are given by frequency-division and can be found using a water-filling algorithm. It is shown that when the transmitters' observations are very noisy our scheme with general parameters outperforms the best previously known schemes. Moreover, the scheme allows to conclude that user cooperation strictly improves the capacity region, irrespective of how noisy the transmitters' observations are. In contrast, when the observations are almost noise-free, then a scheme previously proposed by Carleial improves on our scheme. It is shown that for symmetric setups - i.e., equal power constraints at the two transmitters and equal variances of the noises corrupting the transmitters' observations - and in the asymptotic regime when the observations become noise-free Carleial's lower bound on the sum-capacity and Tandon and Ulukus's recent upper bound on the sum-capacity are tight in the sense that they both tend to the sum-capacity of the setup with noise-free observations with identical slopes.