This article establishes that the capacity region of the two-users weak Gaussian interference channel can be achieved using Gaussian codebooks. It is further shown that the upper concave envelope of the achievable rate region can be attained using at most two time-sharing phases. In one phase, both users transmit simultaneously, while in the second phase, when present, only one user is active. Furthermore, the boundary of the capacity region can be traversed continuously through incremental reallocations of power between the two messages transmitted by each user. Finally, it is proven that the Han-Kobayashi achievable rate region with single-letter Gaussian codebooks attains the optimal boundary of the capacity region.

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