The present invention relates to a guiding sleeve. This sleeve comprises a sleeve holder (2), a sheath (4) secured to the sleeve holder and comprising a grip (3) capable of rotationally guiding a bar (B) to be machined, means for opening and closing the grip (6), and adjusting means capable of controlling the opening and closing means in order to compensate for variations in the dimensional tolerances of the bar (B), this sleeve being characterised in that the adjustment means are capable of actuating the means for opening and closing the grip (6) instantaneously in direct response to the variations in dimensional tolerances of the bar (B), in order to provide a substantially constant pressure of the grip (6) on the bar (B). This sleeve is intended to equip a machine tool such as a cutting- off lathe with a tailstock. IMAGE
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