  We show that the bilinear Hilbert transform $H_{\Gamma}$ along curves
$\Gamma=(t,-\gamma(t))$ with $\gamma\in\mathcal{N}\mathcal{F}^{C}$ is bounded
from $L^{p}(\mathbb{R})\times
L^{q}(\mathbb{R})\,\rightarrow\,L^{r}(\mathbb{R})$ where $p,\,q,\,r$ are
H\"older indices, i.e. $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, with
$1<p<\infty$, $1<q\leq\infty$ and $1\leq r<\infty$. Here
$\mathcal{N}\mathcal{F}^{C}$ stands for a wide class of smooth "non-flat"
curves near zero and infinity whose precise definition is given in Section 2.
This continues author's earlier work on this topic, extending the boundedness
range of $H_{\Gamma}$ to any triple of indices
$(\frac{1}{p},\,\frac{1}{q},\,\frac{1}{r'})$ within the Banach triangle. Our
result is optimal up to end-points.
