The core of a single-core MgB2/Cu,Nb composite, which has been prepared by the ex-situ technique and exhibits a high critical current equal to 427 A (at 0 T and 4.2 K, j (c) a parts per thousand yen 10(5) A/cm(2)), has been studied using various structural methods. Two kinds of MgB2 crystals were observed; those of the first kind is large, highly dense crystals characterized by a low oxygen content (2-8 at %) and the others are fine, weakly coupled crystallites characterized by high oxygen content (4-21 at %). To perform a comparative analysis of the structures, we have also studied an MgB2 bulk sample synthesized at 1000A degrees C. It was found that two phases with the same lattice are formed; they differ in the magnesium and boron contents (within the homogeneity range), impurity oxygen content and microstructure as well but differ slightly in the lattice parameters. The two-phase state of MgB2 bulk sample is due to the mechanism of its formation, which includes the melting of magnesium, the dissolution of solid boron in it, and the crystallization of MgB2 from the melt with the formation of dendrite-like structure characterized by corresponding redistribution of components and impurities. To a certain degree, the two-phase structure of MgB2 bulk sample is inherited by the MgB2/Cu,Nd composite prepared by ex-situ technique (annealing of composite at 700A degrees C). It was shown that oxygen in the MgB2 compound is the destabilizing factor and leads to the transformation of the superconductor into MgO.