The eccentric connectivity index of a connected graph \(G\) is the sum over all vertices \(v\) of the product \(d_G(v)e_G(v)\), where \(d_G(v)\) is the degree of \(v\) in \(G\) and \(e_G(v)\) is the maximum distance between \(v\) and any other vertex of \(G\). This index is helpful for the prediction of biological activities of diverse nature, a molecule being modeled as a graph where atoms are represented by vertices and chemical bonds by edges. We characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order \(n\). Also, given two integers \(n\) and \(p\) with \(p\leq n-1\), we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order \(n\) with \(p\) pending vertices.