Let \(V\) be a normed vector space and \(X\) a Banach space, and let \(f,g,h: V\to X\). The authors prove that the Pexider equation \[ f(x+y)= g(x)+h(y) \] is stable in the following sense: If there exists a real number \(p\neq 1\), such that \[ \bigl\|f(x+y)- g(x)-h(y) \bigr\|\leq\|x \|^p+ \|y\|^p \] for all \(x,y\in V\setminus \{0\}\), then there exists exactly one additive map \(T:V\to X\) such that \[ \bigl\|f(x)- T(x)-f(0) \bigr\|\leq C(p)\|x\|^p \] for all \(x\in V\). Here \(C(p)\) is a certain specified constant.