  The spectrum of edge excitations is derived for the spin-unpolarized $\nu =
2$ and $\nu = 2/3$ FQHE. Numerical diagonalization of a system of six electrons
on a disc confirms that the edge $\nu = 2/3$ spin-singlet FQHE state consists
of oppositely directed spin and charge branches on the same physical edge. The
highly correlated $\nu = 2/3$ singlet edge is shown to have the same spin
branch as the $\nu = 2$ singlet edge, providing evidence that the same
$SU(2)_{k = 1}$ Kac-Moody algebra describes all unmixed spin branches. The spin
and charge branches of the singlet state at $\nu = 2/3$ are less coupled than
the two branches of the spin-polarized state at the same filling factor,
suggesting that the conductance along an edge may increase sharply across the
polarized-unpolarized transition.
