In repeated games, simple strategies such as Grim Trigger, while strict equilibria when monitoring is perfect, can fail to be even approximate Nash equilibria when monitoring is private, yet arbitrarily close to perfect. That is, they fail to be robust to private monitoring. In this paper, it is shown that for a class of repeated Prisoner's Dilemma games these strategies, when viewed as (degenerate) correlated equilibria, are robust. In particular, even when monitoring is private and conditionally independent, as the signalling noise goes to zero, there is a sequence of correlated equilibria converging to the Grim Trigger strategies. The correlation device uses an information structure akin to that of Rubinstein's e-mail game.