  Precision measurements of transitions between singlet ($S=0$) Rydberg states
of H$_2$ belonging to series converging on the
$\mathrm{X}^+\,^2\Sigma_g^+(v^+=0,N^+=0)$ state of H$_2^+$ have been carried
out by millimetre-wave spectroscopy under field-free conditions and in the
presence of weak static electric fields. The Stark effect mixes states with
different values of the orbital-angular-momentum quantum number $\ell$ and
leads to quadratic Stark shifts of low-$\ell$ states and to linear Stark shifts
of the nearly degenerate manifold of high-$\ell$ states. Transitions to the
Stark manifold were observed for the principal numbers 50 and 70, at fields
below 50 mV/cm, with linewidths below 500~kHz. The energy-level structure was
calculated using a matrix-diagonalisation approach, in which the zero-field
positions of the $\ell\leq 3$ Rydberg states were obtained either from
multichannel-quantum-defect-theory calculations or experiment, and those of the
$\ell\geq 4$ Rydberg states from a long-range core-polarisation model. This
approach offers the advantage of including rovibronic channel interactions
through the MQDT treatment while retaining the advantages of a spherical basis
for the determination of the off-diagonal elements of the Stark operator.
Comparison of experimental and calculated transition frequencies enabled the
quantitative description of the Stark manifolds, with residuals typically below
50 kHz. We demonstrate how the procedure leads to quantum defects and binding
energies of high Rydberg states with unprecedented accuracy, opening up new
prospects for the determination of ionisation energies in molecules.
