Let \(X\) be a holomorphic vector field on a compact Kähler manifold \((M,\omega)\). Then \(\omega\) is a Kähler-Ricci soliton with respect to \(X\) if \(\text{Ric} (\omega)-\omega=L_X\omega\) where \(\text{Ric} (\omega)\) is the Ricci form and \(L_X\) is the Lie derivative with respect to \(X\). It is easily seen that in this case \(c_1(M)>0\) and that \(c_1(M)\) is represented by \(\omega\). On the other hand, by computing the Futaki invariant one finds that if a Kähler-Ricci soliton exists, then the manifold cannot admit any Kähler-Einstein metrics. The paper under review proves the uniqueness (modulo a certain subgroup of Kähler automorphisms) of a Kähler-Ricci soliton on a fixed compact Kähler manifold. The proof consists of discussing a Monge-Ampère equation depending on a parameter \(t\in [0,1]\) such that for \(t=1\) one obtains the Kähler-Ricci soliton equation.