We have investigated the magnetic, thermodynamic, transport, and electronic properties across the transition from the divalent antiferromagnetic state to the valence fluctuating state of Eu in the alloy EuCu2(Ge1-xSix)(2). The antiferromagnetic state is very stable from x = 0 up to x = 0.6 and disappears rather abruptly at x(c) approximate to 0.65. Near the crossover, we confirmed a pronounced Kondo-like behavior in the resistivity and in the thermopower. Further on, for x slightly larger than x(c), we observe the formation of a heavy Fermi liquid at low temperatures, as evidenced by a large linear coefficient of the specific heat (gamma = 191 mJ/K-2 mol for x = 0.7), a large quadratic term in the resistivity and a strongly enhanced constant susceptibility chi(0). This is a unique observation of heavy fermion behavior in a Eu compound. The photoemission spectra of the Eu 4f and 3d core levels indicate significant valence fluctuation even for x<x(c) in the magnetically ordered regime. The decrease of T-N before reaching the critical point as well as the Kondo and the heavy fermion behavior found near x(c) are in strong contrast to the observation in typical Eu systems, which usually show a first-order transition from a (almost) divalent state to a strongly valence fluctuating Eu state.