  We compute the masses of the $\pi$ and of the $\rho$ mesons in the quenched
approximation on a lattice with fixed lattice spacing $a \simeq 0.145 \
\mathrm{fm}$ for SU($N$) gauge theory with $N = 2,3,4,6$. We find that a simple
linear expression in $1/N^2$ correctly captures the features of the
lowest-lying meson states at those values of $N$. This enables us to
extrapolate to $N = \infty$ the behaviour of $m_{\pi}$ as a function of the
quark mass and of $m_{\rho}$ as a function of $m_{\pi}$. Our results for the
latter agree within 5% with recent predictions obtained in the AdS/CFT
framework.
