Sparse random graphs with prescribed degree distribution have attracted lot of attention thanks to their relevance to web modeling. Molloy and Reed [MR98] have characterized those limiting degree distributions for which the corresponding random graphs contained with overwhelming probability a giant connected component. We complement their law of large numbers by a central limit theorem: the limiting size of the giant component of random graphs with given degree distribution is Gaussian. The proof revisits the analysis of the fluctuations of the giant component for Erdos-Rényi graphs carried out in Barraez, Boucheron and Fernandez de la Vega [BBFdlV00]. This is a martingale-oriented analysis of a graph exploration.