We identify room-temperature converse magnetoelectric effects (CMEs) that are non-volatile by using a single-crystal substrate of PMN–PT (001)pc (pc denotes pseudocubic) to impart voltage-driven strain to a polycrystalline film of Ni. An appropriate magnetic-field history enhances the magnetoelectric coefficient to a near-record peak of ∼10−6 s m−1 and permits electrically driven magnetization reversal of substantial net magnetization. In zero magnetic field, electrically driven ferroelectric domain switching produces large changes of in-plane magnetization that are non-volatile. Microscopically, these changes are accompanied by the creation and destruction of magnetic stripe domains, implying the electrical control of perpendicular magnetic anisotropy. Moreover, the stripe direction can be rotated by a magnetic field or an electric field, the latter yielding the first example of electrically driven rotatable magnetic anisotropy. The observed CMEs are associated with repeatable ferroelectric domain switching that yields a memory effect. This memory effect is well known for PMN–PT (110)pc but not PMN–PT (001)pc. Given that close control of the applied field is not required as for PMN–PT (110)pc, this memory effect could lead the way to magnetoelectric memories based on PMN–PT (001)pc membranes that switch at low voltage.