A machine for producing parts made of composite material, the machine comprising a fiber-application head comprising an application roller, a guide system for guiding one or more fibers, in the form of a strip, towards said application roller, and at least one cutting system comprising a blade capable of being maneuvered by a cutting cylinder between a rest position and an active position for cutting a fiber, each cutting system comprising a counter-blade that is elastically held against the blade by the cutting edge thereof for cutting by shearing a fiber while the blade moves from the rest position thereof to the active position thereof.