  In this paper, we prove that the Cauchy problem associated to the following
higher-order Benjamin-Ono equation $$ \partial_tv-b\mathcal{H}\partial^2_xv-
a\epsilon \partial_x^3v=cv\partial_xv-d\epsilon
\partial_x(v\mathcal{H}\partial_xv+\mathcal{H}(v\partial_xv)), $$ is globally
well-posed in the energy space $H^1(\mathbb R)$. Moreover, we study the limit
behavior when the small positive parameter $\epsilon$ tends to zero and show
that, under a condition on the coefficients $a$, $b$, $c$ and $d$, the solution
$v_{\epsilon}$ to this equation converges to the corresponding solution of the
Benjamin-Ono equation.
