A MEMS device wherein a die of semiconductor material has a first face and a second face. A membrane is formed in or on the die and faces the first surface. A cap is fixed to the first face of the first die and is spaced apart from the membrane by a space. The die is fixed, on its second face, to an ASIC, which integrates a circuit for processing the signals generated by the die. The ASIC is in turn fixed on a support. A packaging region coats the die, the cap, and the ASIC and seals them from the outside environment. A fluidic path is formed through the support, the ASIC, and the first die, and connects the membrane and the first face of the die with the outside, without requiring holes in the cap.