The cryogenic targets of the Laser Megajoule facility (LMJ) are hollow spheres. Their internal walls are covered with a solid layer of frozen deuterium-tritium (D-T). One issue of inertial confinement fusion experiments is to guarantee the quality of the geometry of fuel layer. Cryogenic targets must be cooled at a temperature near the triple point (19K) with a very good stability (0.2mK) for many hours. This period is used to position the target with an accuracy of ±5μm at the center of the experimental vacuum vessel where the 240 laser beams are focalized. A complex cryogenic infrastructure has been conceived to insure the continuity of the cryogenic chain from the filling station located at CEA/Valduc in Burgundy to the LMJ experimental chamber installed in the vicinity of Bordeaux. The design of the target and a detailed description of the infrastructure are presented. A first prototype of cryogenic grip has been fabricated and characterized. Some experimental results are given.