Summary: We prove using the Fourier restriction norm method that the initial value problem associated to a perturbation of the Benjamin-Ono equation \(u_t+uu_x+{\beta}\mathcal Hu_{xx}+{\eta}(\mathcal Hu_x-u_{xx})=0\), where \(x\in \mathbb R\), \(t\geq 0\), \({\eta}>0\) and \(\mathcal H\) denotes the usual Hilbert transform, is locally well-posed in the Sobolev space \(H^s(\mathbb R)\) for any \(s>-1/2\) and globally well-posed in \(H^s(\mathbb R)\) when \(s\geq 0\). For \(s<-1\) we also prove some ill-posedness issues.