The Karp-Sipser algorithm consists in removing recursively the leaves as well their unique neighbours and all isolated vertices of a given graph. The remaining graph obtained when there is no leaf left is called the Karp-Sipser core. When the underlying graph is the classical sparse Erdős-Rényi random graph G[n, c/n], it is known to exhibit a phase transition at c = e. We show that at criticality, the Karp-Sipser core has size of order n ^{3/5} , which proves a conjecture of Bauer and Golinelli. We provide the asymptotic law of this renormalized size as well as a description of the distribution of the core as a graph. Our approach relies on the differential equation method, and builds up on a previous work on a configuration model with bounded degrees.