The precise determination of the Lamb shift in heavy hydrogen-like ions provides a sensitive test of quantum electrodynamics in very strong Coulomb fields, not accessible otherwise. For the investigation of the Lyman-alpha transitions in Pb-208(81+) Or U-238(91+) With sufficient accuracy a high resolving calorimetric detector for hard X-rays (E less than or equal to 100 keV) is presently developed. The detector modules consist of arrays of silicon thermistors and of X-ray absorbers made of high Z material to optimize the absorption efficiency. The detectors are housed in a specially designed He-3/He-4 dilution refrigerator with a side arm which fits to the geometry of the internal target of the storage ring ESR at GSI Darmstadt. The detector performance presently achieved is already close to fulfill the demands of the Lamb shift experiment. For a prototype detector pixel with a 0.3 mm(2) x 66 mu m Sn absorber an energy resolution of Delta E-FWHM = 75 eV is obtained for 60 keV X-rays. (C) 2000 Elsevier Science B.V. All rights reserved.