The described process for making a continuous composite material structure (50) starts from a continuous semifinished product (1) made of a reinforcing fiber comprising an upper layer (4) and a lower layer (5) facing each other and mutually connected by intermediate threads (6). It comprises the steps of moving the continuous semifinished product (1) along a working path; impregnating the semifinished product (1) with a resin; coupling a lower surface of the continuous semifinished product (1) with a first ribbon (23); moving one of said layers (4, 5) apart from the opposite layer for bringing the intermediate threads (6) from a first condition in which they are disposed substantially parallel to the two external layers (4, 5) to a second condition in which they are disposed transversally of the external layers; and polymerizing the impregnated semifinished product (1) to make the continuous structure (50).