This study reports reliable theoretical evidence of the existence of a new diatomic molecule, the species BeB. Complementing previous investigations predicting also the existence of new species like : BeN and BeC, it allows one to have now a global view of the properties and bonding of diatomics of beryllium with first/second row atoms. The electronic structure was studied at the multireference single and double excitations configuration interaction (MRSDCI) level as implemented in the MELD codes and with natural orbitals (for the (4) Sigma(-) state) as the one-particle basis. Eighteen electronic states have been characterized with the ground state being a X(2)II with a relatively long equilibrium internuclear distance (R(e) = 3.654 a(0)) and a small dissociation energy (D-e = 0.86 eV). The next doubler is also weakly bound with R(e) = 3.982 a(0) and D-e = 0.48 eV. These states are followed by two close-lying quarters at 0.568 eV (a(4) Sigma(-)), R(e) = 3.246 a(0), and at 0.593 eV (b(4)II), R(e) = 3.460 a(0), with much deeper potentials, 3.04 and 3.02 eV, respectively. The curves crossings between these three excited states and another group of four doubler states lying : between 1.5 and 1.8 eV will certainly be responsible for a complex structure in the spectra of this molecule. Complementing the electronic description of all states, vibrational and rotational constants are also presented as well as dipole moment functions and vibrationally averaged dipole moments. A graphical display of contour maps of the valence orbitals as a function of the internuclear distance clearly illustrates the electron rearrangements occurring during molecule formation and the participation of beryllium 2p orbital in a pi-type bond. BeB shows more resemblance with the isoelectronic molecule B-2(+) rather than with BeC+. A comparison o,f the ground state spectroscopic properties of BeB with the other first/second row beryllium diatomics is also presented.