  In this article, the open problem of finding the exact value of the norm of
the Hilbert matrix operator on weighted Bergman spaces $A^p_\alpha$ is
adressed. The norm was conjectured to be $\frac{\pi}{\sin
\frac{(2+\alpha)\pi}{p}}$ by Karapetrovi\'{c}. We obtain a complete solution to
the conjecture for $\alpha \ge 0$ and
$2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3} \le p < 2(2+\alpha)$ and a
partial solution for $2+2\alpha < p <
2+\alpha+\sqrt{\alpha^2+\frac{7}{2}\alpha+3}.$ Moreover, we also show that the
conjecture is valid for small values of $\alpha$ when $2+2\alpha < p \le
3+2\alpha.$ Finally, the case $\alpha = 1$ is considered.
