This invention relates to a cutting unit for a well pipe. The cutting unit includes a tool housing axially displaceable and rotatable in the well pipe, arranged in the tool housing at least one cutting tool displaceable in a radial direction, the unit connected to a rotary adapter for rotation around a center axis, and the tool housing provided with a leading end portion and a coupling portion, the leading end portion being conical, sloping towards the center axis and away from the coupling portion; and the cutting unit provided with an elongated springable tool mount with a first end portion, a second end portion and a resilient portion between first end portion and second end portion, elongated tool mount positioned in an external recess in the tool housing, and the second end portion positioned at the leading end portion, and the tool mount tensioned towards the center axis; a sleeve positioned around the coupling portion, the first end portion attached to the sleeve; at least one shear body fixing the sleeve to the tool housing; an internal first bevel on the second end portion of the tool mount; a cutting unit attached to the second end portion on the opposite side of the internal first bevel; and a push rod connected to a cone body, cone body resting against the first bevel.