We describe an algorithm to compute the integral closure Ī of an ideal I in a polynomial ring R:=k[x₁,…,x_n] over a field. It is based on the known fact that Ī is encoded in the integral closure of the Rees algebra [Rscr ](I):=⌖_k≥0I^kt^k⊆R[t] of I and on an algorithm to compute the normalization of an affine domain. In particular we explain an implementation of this algorithm in the computer algebra system SINGULAR [11]. As an application we mention the connection of the integral closure of ideals to the study of Whitney equisingularity.