  Polar molecules are desirable systems for quantum simulations and cold
chemistry. Molecular ions are easily trapped, but a bias electric field applied
to polarize them tends to accelerate them out of the trap. We present a general
solution to this issue by rotating the bias field slowly enough for the
molecular polarization axis to follow but rapidly enough for the ions to stay
trapped. We demonstrate Ramsey spectroscopy between Stark-Zeeman sublevels in
180Hf19F+ with a coherence time of 100 ms. Frequency shifts arising from
well-controlled topological (Berry) phases are used to determine magnetic
g-factors. The rotating-bias-field technique may enable using trapped polar
molecules for precision measurement and quantum information science, including
the search for an electron electric dipole moment.
