  The dramatic slowing down associated with the glass transition cannot be
fully understood without an associated static length that is expected to
increase rapidly as the temperature is reduced. The search for such a length
was long and arduous, without a universally accepted candidate at hand.
Recently a natural such length $\xi_s$ was proposed, stemming from a cross-over
between plastic and elastic mechanical responses of the material. In this
Letter we show that supercooled liquids in which there exists random pinning
sites of density $\rho_{\rm im}\sim 1/\xi_s^d$ exhibit complete jamming of all
dynamics. This is a direct demonstration that the proposed length scale is
indeed {\em the} static length that was long sought-after.
