A device for supplying a liquid cooling lubricant to a cutting edge of a lathe tool is operatively connected with a fast-tool servo by way of a tool holder in order to be able to execute oscillating movements in an advance direction in common with the tool holder. A first connection is stationary with respect to a housing of the fast-tool servo, for a cooling lubricant source and an outlet, which is in fluid connection with the first connection, for delivery of a cooling lubricant jet to the lathe tool. The outlet is provided at the tool holder and is in fluid connection with the first connection by way of a hose portion so that the cooling lubricant jet delivered via the outlet always impinges at a predetermined point on the cutting edge of the lathe tool even when stroke movements of the tool holder take place.