Communication systems suffer from the mixed noise consisting of both non-Gaussian impulsive noise (IN) and white Gaussian noise (WGN) in many practical applications. However, there is little literature about the channel capacity under mixed noise. In this paper, we prove the existence of the capacity under p-th moment constraint and show that there are only finite mass points in the capacity-achieving distribution. Moreover, we provide lower and upper capacity bounds with closed forms. It is shown that the lower bounds can degenerate to the well-known Shannon formula under special scenarios. In addition, the capacity for specific modulations and the corresponding lower bounds are discussed. Numerical results reveal that the capacity decreases when the impulsiveness of the mixed noise becomes dominant and the obtained capacity bounds are shown to be very tight.