A process and an apparatus for cutting a continuous rubber-covered belt in a process for building tyres for vehicle wheels, wherein the apparatus includes a cutting group, in turn comprising: a knife formed by a first half-part and a second half-part and movable along a cutting direction that is tilted with respect to a supply direction of a rubber-covered belt placed beneath the knife. The process includes: lifting a lifting element placed below the continuous rubber-covered belt and at the cutting zone, until a portion of rubber-covered belt is lifted, and lowering the single first-half part of the knife, making it slide with respect to the second half-part.