In an installation of a wind turbine on a floating foundation in floating condition and subject to sea-state induced motions, e.g. at the site of an offshore windfarm, use is made of a vessel with a crane arranged on the hull and provided with a hoisting system adapted to support the weight of the wind turbine and suspend the wind turbine from the crane. A heave compensation device compensates for sea-state induced heave motion of the wind turbine mast relative to the mast mounting structure of the floating foundation. Use is made of a mast alignment system configured to engage on the suspended wind turbine, e.g. on the mast of the suspended wind turbine, and to bring and maintain the mast of the wind turbine in alignment with the mounting axis of the floating foundation in order to compensate for sea-state induced motions, at least including tilt motions in one or more vertical planes, of the wind turbine mast relative to the mounting axis of the floating foundation.