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,