  Cs$_2$ Rydberg-ground molecules consisting of a Rydberg, $nD_{J}$ (33 $\leq$
$n$ $\leq$ 39), and a ground state atom, 6$S_{1/2} (F=$3 or 4$)$, are
investigated by photo-association spectroscopy in a cold atomic gas. We observe
vibrational spectra that correspond to triplet $^T\Sigma$ and mixed
$^{S,T}\Sigma$ molecular states. We establish scaling laws for the energies of
the lowest vibrational states vs principal quantum number and obtain
zero-energy singlet and triplet $s$-wave scattering lengths from experimental
data and a Fermi model. Line broadening in electric fields reveals the
permanent molecular electric-dipole moments; measured values agree well with
calculations. We discuss the negative polarity of the dipole moments, which
differs from previously reported cases.
