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, i.e. an electric field of 5 kV/cm, we predict a large magnetic dipole moment up to 13 Bohr magnetons, and sizeable electric dipole moment up to 0.22 debye. When a magnetic field is present, we show that the induced 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.