  A system of hard rigid rods of length $k$ on hypercubic lattices is known to
undergo two phases transitions when chemical potential is increased: from a low
density isotropic phase to an intermediate density nematic phase, and on
further increase to a high-density phase with no orientational order. In this
paper, we argue that, for large $k$, the second phase transition is a first
order transition with a discontinuity in density in all dimensions greater than
$1$. We show the chemical potential at the transition is $\approx k \ln [k /\ln
k]$ for large $k$, and that the density of uncovered sites drops from a value $
\approx (\ln k)/k^2$ to a value of order $\exp(-ak)$, where $a$ is some
constant, across the transition. We conjecture that these results are
asymptotically exact, in all dimensions $d\geq 2$. We also present evidence of
coexistence of nematic and disordered phases from Monte Carlo simulations for
rods of length $9$ on the square lattice.
