A new method for the preparation of single crystals of the superconducting intermetallic MgB 2 compound from a Mg-Cu-B melt flux is presented. The high vapour pressure of Mg at elevated temperature is a serious challenge of the preparation process. The approximate thermodynamic calculations of the ternary Mg-Cu-B phase diagram show a beneficial effect of Cu, which extends the range of formation of MgB 2 to lower temperatures. Within the as-solidified Mg-Cu-B melt flux the MgB 2 compound forms plate-like single crystals up to a size of 0.2x0.2x0.05 mm 3 or alternatively rims peritectically grown around MgB 4 particles. AC-susceptibility measurements were conducted with specimens selected from different parts of the as-solidified flux containing MgB 2 particles. Peritectically formed MgB 2 -particles display the highest transition temperature of T c =39.2 K and a relatively narrow transition width of ΔT c =1.3 K. Other sections of the sample exhibit various superconducting transitions from T c =39 K to 7.2 K. This variation of T c is attributed to a finite homogeneity range of the MgB 2 compound whereas significant Cu solid solubility in MgB 2 can be excluded.