Given a continuous time Markov Chain ${q(x,y)}$ on a finite set $S$, the associated noisy voter model is the continuous time Markov chain on ${0,1}^S$ which evolves by (1) for each two sites $x$ and $y$ in $S$, the state at site $x$ changes to the value of the state at site $y$ at rate $q(x,y)$ and (2) each site rerandomizes its state at rate 1. We show that if there is a uniform bound on the rates ${q(x,y)}$ and the corresponding stationary distributions are “almost” uniform, then the mixing time has a sharp cutoff at time $\log |S|/2$ with a window of order 1. Lubetzky and Sly proved cutoff with a window of order 1 for the stochastic Ising model on toroids: we obtain the special case of their result for the cycle as a consequence of our result.