An apparatus for injecting an accelerating fluid into an oil well ; said apparatus comprises a casing which is open at the ends into which a cement flows, said casing to be positioned inside the well forming an annular zone with its walls; said casing comprising in its inside a hollow cylindrical body whose lateral surfaces are integral with the internal surfaces of the casing ; said hollow cylindrical body being characterized by : - having a central hole through which a cement flows, with a lower horizontal section than the casing section, open at both ends and as long as the cylindrical body ; - comprising a hollow internal chamber which contains an accelerating fluid; - having one or more body inlet devices , which put said casing in fluid communication with said chamber ; said inlet devices being positioned on one of the base surfaces of the hollow cylindrical body ; - having one or more body outlet devices , which put said chamber in fluid communication with a conduit through the central hole , - positioned radially with respect to the axis of said cylindrical body ; - positioned along the internal surface of the central hole ; so that the accelerating fluid enters the hole through said devices and mixes with the drilling cement; - means for opening said chamber inlet and outlet devices , wherein mechanical devices are present inside the hollow cylindrical chamber which have the function of slowing down the leakage of the accelerating fluid. said hollow cylindrical body being characterized by : - having a central hole through which a cement flows, with a lower horizontal section than the casing section, open at both ends and as long as the cylindrical body ; - comprising a hollow internal chamber which contains an accelerating fluid; - having one or more body inlet devices , which put said casing in fluid communication with said chamber ; said inlet devices being positioned on one of the base surfaces of the hollow cylindrical body ; - having one or more body outlet devices , which put said chamber in fluid communication with a conduit through the central hole , - positioned radially with respect to the axis of said cylindrical body ; - positioned along the internal surface of the central hole ; - having a central hole through which a cement flows, with a lower horizontal section than the casing section, open at both ends and as long as the cylindrical body ; - comprising a hollow internal chamber which contains an accelerating fluid; - having one or more body inlet devices , which put said casing in fluid communication with said chamber ; said inlet devices being positioned on one of the base surfaces of the hollow cylindrical body ; - having one or more body outlet devices , which put said chamber in fluid communication with a conduit through the central hole , - positioned radially with respect to the axis of said cylindrical body ; - positioned along the internal surface of the central hole ; - positioned radially with respect to the axis of said cylindrical body ; - positioned along the internal surface of the central hole ; so that the accelerating fluid enters the hole through said devices and mixes with the drilling cement; - means for opening said chamber inlet and outlet devices , - means for opening said chamber inlet and outlet devices , <EOS>