The author studies the Hyers-Ulam-Rassias stability of a Jensen type functional equation \[ 3f((x+y+z)/3)+ f(x)+ f(y)+ f(z)= 2[ f((x+y)/2)+ f((y+z)/2)+ f((z+x)/2)]. \] The main result of this paper is the following: If the function \(f: X\to Y\) satisfies \[ \begin{multlined}\|3 f((x+y+z)/3)+ f(x)+ f(y)+ f(z)- 2[f((x+y)/2)+ f((y+z)/2)+ f((z+x)/2)]\|\\ \leq \delta + \theta (\|x\|^p+ \|y\|^p+ \|z\|^p)\end{multlined} \] for all \(x,y,z\in X\), then there is a unique additive mapping \(A: X\to Y\) such that \[ \|f(x)- f(0)- A(x)\|\leq {{\delta}\over 3}+ {{\theta}\over {2^{1-p}-1}} \|x\|^p \] for all \(x\in X\). Here \(X\) and \(Y\) are real normed linear spaces, and \(\delta,\theta\in [0, \infty)\) and \(p\in(0,1)\).