Let \(Hf(x, y):= f(x+ y- xy)+ f(xy)- f(x)- f(y)\). The following result on Hyers-Ulam stability of the Hosszú equation \(Hf(x, y)= 0\) is proved: Let \(f: \mathbb{R}\to \mathbb{R}\) be a function satisfying \(|Hf(x, y)|\leq \delta\) for some \(\delta> 0\). There exists an additive function \(a: \mathbb{R}\to \mathbb{R}\) such that the difference \(f- a\) is bounded iff the even part \(h\) of \(f\) satisfies \(|Hh(x, y)|\leq \varepsilon\) for some \(\varepsilon> 0\).