This article shows that the capacity region of a two users weak Gaussian interference channel can be achieved using single letter Gaussian code-books. It is shown that the upper concave envelope of the capacity region can be achieved using at most two time-sharing phases. In one phase, both users are active, while in the second phase, if it exists, only a single user is active. It is shown that the boundary of the capacity region can be traversed continuously through incremental reallocation of power between the messages of each user. Finally, it is established that the solution to the Han-Kobayashi achievable rate region, with single letter Gaussian code-books, achieves the optimum boundary.