  In a previous letter, we computed the decay constants of heavy pseudoscalar
mesons in the framework of relativistic (instantaneous) Bethe-Salpeter method
(full $0^-$ Salpeter equation), in this letter, we solve the full $1^-$
Salpeter equation and compute the leptonic decay constants of heavy-heavy and
heavy-light vector mesons. The theoretical estimate of mass spectra of these
heavy-heavy and heavy-light vector mesons are also presented. Our results for
the decay constants and mass spectra include the complete relativistic
contributions. We find $F_{D^*_s} \approx 375 \pm 24 $, $F_{D^*} \approx 340
\pm 23 (D^{*0},D^{*\pm})$, $F_{B^*_s} \approx 272 \pm 20 $, $F_{B^*} \approx
238 \pm 18 (B^{*0},B^{*\pm})$, $F_{B^*_c} \approx 418 \pm 24 $, $F_{J/\Psi}
\approx 459 \pm 28 $, $F_{\Psi(2S)} \approx 364 \pm 24 $, $F_{\Upsilon} \approx
498 \pm 20 $ and $F_{\Upsilon(2S)} \approx 366 \pm 27 $ MeV.
