  We have proposed in a series of previous papers a method to determine the
effective axial-vector current coupling and the strength of the isoscalar
proton-neutron pairing interaction for calculating the nuclear matrix elements
of the neutrinoless double-$\beta$ decay by the quasiparticle random-phase
approximation. The combination of these two parameters have had an uncertainty
in this approach, but now this uncertainty is removed. In this paper, we apply
our method to the neutrinoless double-$\beta$ decays of $^{136}$Xe and
$^{130}$Te and predict the nuclear matrix elements and reduced half-lives. Our
calculation is tested first by a self-check method using the two-neutrino
double-$\beta$ decay, and this test ensures the application of our method to
$^{136}$Xe. It turns out, however, that our method is not successful in
$^{130}$Te. Further test is made for our calculation of the $\beta$ decay of
$^{138}$Xe, and a satisfactory result is obtained.
