  Graphene supported on a transition metal dichalcogenide substrate offers a
novel platform to study the spin transport in graphene in presence of a
substrate induced spin-orbit coupling, while preserving its intrinsic charge
transport properties. We report the first non-local spin transport measurements
in graphene completely supported on a 3.5 nm thick tungsten disulfide (WS$_2$)
substrate, and encapsulated from the top with a 8 nm thick hexagonal boron
nitride layer. For graphene, having mobility up to 16,000
cm$^2$V$^{-1}$s$^{-1}$, we measure almost constant spin-signals both in
electron and hole-doped regimes, independent of the conducting state of the
underlying WS$_2$ substrate, which rules out the role of spin-absorption by
WS$_2$. The spin-relaxation time $\tau_{\text{s}}$ for the electrons in
graphene-on-WS$_2$ is drastically reduced down to~10 ps than $\tau_{\text{s}}$
~ 800 ps in graphene-on-SiO$_2$ on the same chip. The strong suppression of
$\tau_{\text{s}}$ along with a detectable weak anti-localization signature in
the quantum magneto-resistance measurements is a clear effect of the WS$_2$
induced spin-orbit coupling (SOC) in graphene. Via the top-gate voltage
application in the encapsulated region, we modulate the electric field by 1
V/nm, changing $\tau_{\text{s}}$ almost by a factor of four which suggests the
electric-field control of the in-plane Rashba SOC. Further, via carrier-density
dependence of $\tau_{\text{s}}$ we also identify the fingerprints of the
D'yakonov-Perel' type mechanism in the hole-doped regime at the graphene-WS$_2$
interface.
