[en] The dynamics of a quantum particle tunneling through the barrier of an almost symmetric double-well potential is studied. The coherent quantum oscillations characteristic for isolated tunneling systems are shown to be strongly affected by the interaction with a heat bath environment. Attention is focused on the situation where the heat bath causes an Ohmic damping. This case is of potential relevance to the problem of macroscopic quantum coherence in SQUID rings. The range of parameters in which quantum coherence survives is determined and the time evolution of the occupation probabilities in the two wells is calculated