Highly sensitive imaging observations of the Galactic center (GC) at high energies with an angular resolution of order 10 arcminutes, is a very recent development in the field of high-energy astrophysics. The Ibis/Isgri imager on the Integral observatory detected for the first time a hard X-ray source, IGR, located within 1 arcminute of Sagittarius A* (Sgr A*) over the energy range 20-100 keV. Here we present the results of a detailed analysis of approximately 7 * 106 s of observations of the GC obtained since the launch of Integral in October 2002. Two years and an effective exposure of 4.7 * 106 s have allowed us to obtain more stringent positional constraints on this high-energy source and to construct its spectrum in the range 20-400 keV. Furthermore, by combining the Isgri spectrum with the total X-ray spectrum corresponding to the same physical region around Sgr A* from XMM data, and collected during part of the gamma-ray observations, we constructed and present the first accurate wide band high-energy spectrum for the central arcminutes of the Galaxy. Our complete and updated analysis of the emission properties of the Integral source shows that it is faint but persistent with no variability above 3 sigma , contrary to what was alluded to in our first paper. This result, in conjunction with the spectral characteristics of the soft and hard X-ray emission from this region, suggests that the source is most likely not point-like but, rather, that it is a compact, yet diffuse, non-thermal emission region. The centroid of IGR is estimated to be R.A. = 17h45m42s.5, decl. = -28o59'28'' (J2000), offset by 1' from the radio position of Sgr A* and with a positional uncertainty of 1'. Its 20-400 keV luminosity at 8 kpc is L = (5.37 +/- 0.21) * 1035 erg/s. A 3 sigma upper limit on the flux at the electron-positron annihilation energy of 511 keV from the direction of Sgr A* is set at 1.9 * 10-4 ph cm-2 s-1. Very recently, the Hess collaboration presented the detection of a source of TeV gamma-rays also located within an arcminute of Sgr A*. We present arguments in favor of an interpretation according to which the photons detected by Integral and Hess arise from the same compact region of diffuse emission near the central black hole and that the supernova remnant Sgr A East could play an important role as a contributor of very high-energy gamma-rays to the overall spectrum from this region. There is also evidence for hard emission from a region located between the central black hole and the radio Arc near l 0.1o along the Galactic plane and known to contain giant molecular clouds.