We consider the preferential attachment model. This is a growing random graph such that at each step a new vertex is added and forms $m$ connections. The neighbors of the new vertex are chosen at random with probability proportional to their degree. It is well known that the proportion of nodes with a given degree at step $n$ converges to a constant as $n\rightarrow\infty$. The goal of this paper is to investigate the asymptotic distribution of the fluctuations around this limiting value. We prove a central limit theorem for the joint distribution of all degree counts. In particular, we give an explicit expression for the asymptotic covariance. This expression is rather complex, so we compute it numerically for various parameter choices. We also use numerical simulations to argue that the convergence is quite fast. The proof relies on the careful construction of an appropriate martingale.