In a process employing a coffee machine, a sealed capsule is fed with a certain amount of axial clearance into a cup aligned along a horizontal axis with a pressurized-hot-water sprinkler, and is moved towards the sprinkler to press an annular flange of the sealed capsule in fluidtight manner against the sprinkler, to pierce a front sealing wall of the sealed capsule by means of a first piercing device, to feed pressurized hot water through the sealing wall to expand the sealed capsule axially and eliminate the axial clearance, and to bring an end wall of the sealed capsule into fluidtight contact with a second piercing device, and pierce the end wall against the second piercing device, so that coffee flows out through the second piercing device.