A device (1) for making a cut in a tissue, such as a blood vessel (2), capsule and the like, includes a trocar (3), the distal end (4) of which can be placed on the tissue (2) and inside which trocar (3) a reduced pressure can be generated for holding the tissue pressed against the distal end (3) under the influence of the reduced pressure, as well as a cutting member (6) located in the trocar (3), such that the tissue (2) pressed against the distal end (4) and the cutting member (6) can be moved relative to one another to make the cut (9). A tissue support (10) is provided in the trocar (3) close to the distal end (4) thereof with which the tissue (2) that extends over the lumen of the trocar (3) can make contact under the influence of the reduced pressure inside the trocar (3).