We analyze infinitely repeated prisoners' dilemma games with imperfect private monitoring, and construct sequential equilibria where strategies are measurable with respect to players' beliefs regarding their opponents' continuation strategies. We show that, when monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of prisoner dilemma games. We also extend the approximate efficiency result to n-player prisoners' dilemma games and to prisoner's dilemma games with more general information structure. Our results require that monitoring be sufficiently accurate but do not require very low discounting.