  In this paper, we prove that the product (in the distribution sense) of two
functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be
written as the sum of two continuous bilinear operators, one from
$\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from
$\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted
by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set
of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies
$$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal
Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of
paraproducts. As a consequence, we find an endpoint estimate involving the
space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.
