We analyze the channel capacity of a system with a large number of one-bit transceivers in a classical Rayleigh environment with perfect channel information at the receiver. With $M$ transmitters and $N=αM$ receivers, we derive an expression of the capacity per transmitter $\mathcal{C}$, where $\mathcal{C}\leq\min(1,α)$, as a function of $α$ and signal-to-noise ratio (SNR) $ρ$, when $M\to\infty$. We show that our expression is a good approximation for small $M$, and provide simple approximations of $\mathcal{C}$ for various ranges of $α$ and SNR $ρ$. We conclude that at high SNR, $\mathcal{C}$ reaches its upper limit of one only if the ratio $α>1.24$. Expressions for determining when $\mathcal{C}$ "saturates" as a function of $α$ and $ρ$ are given.