Previous results suggest that the crosstalk produced by the fiber nonlinearity in a WDM system imposes a severe limit to the capacity of optical fiber channels, since the interference power increases faster than the signal power, thereby limiting the maximum achievable signal-to-interference-plus-noise ratio (SINR). We study this system in the weakly nonlinear regime as a multiple-access channel, and show that by optimally using the information from all the channels for detection, the change in the capacity region due to the nonlinear effect is minimal. On the other hand, if the receiver uses the output of only one wavelength channel, the capacity is significantly reduced due to the nonlinearity, and saturates as the interference power becomes comparable to the noise, which is consistent with earlier results. The results hold in channels with or without memory. Every point in the capacity region can be achieved without knowledge of the nonlinearity parameters at the transmitters. The structures of optimal/suboptimal receivers are briefly discussed