We extend a result by Füredi and Komlós and show that the first eigenvalue of a random graph is asymptotically normal, both for $G_{n,p}$ and $G_{n,m}$, provided $np\geq n^\delta$ or $m/n\geq n^\delta$ for some $\delta>0$. The asymptotic variance is of order $p$ for $G_{n,p}$, and $n^{-1}$ for $G_{n,m}$. This gives a (partial) solution to a problem raised by Krivelevich and Sudakov.

The formula for the asymptotic mean involves a mysterious power series.