  In this paper, we study the non-vanishing of the central values of the
Rankin-Selberg $L$-function of two ad\`elic Hilbert primitive forms ${\bf f}$
and ${\bf g}$, both of which have varying weight parameter $k$. We prove that,
for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$
ad\`elic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12,
{\bf f}\otimes{\bf g})$ are nonzero.
