  We propose a new method to produce an electric and magnetic dipolar gas of
ultracold dysprosium atoms. The pair of nearly degenerate energy levels of
opposite parity, at 17513.33 cm$^{-1}$ with electronic angular momentum $J=10$,
and at 17514.50 cm$^{-1}$ with $J=9$, can be mixed with an external electric
field, thus inducing an electric dipole moment in the laboratory frame. For
field amplitudes relevant to current-day experiments, we predict a magnetic
dipole moment up to 13 Bohr magnetons, and an electric dipole moment up to 0.22
Debye, which is similar to the values obtained for alkali-metal diatomics. When
a magnetic field is present, we show that the electric dipole moment is
strongly dependent on the angle between the fields. The lifetime of the
field-mixed levels is found in the millisecond range, thus allowing for
suitable experimental detection and manipulation.
