Hart will replace Neil Warnock, who took over at Queens Park Rangers on Tuesday.
Hart left Portsmouth in November and joined QPR the next month after only five games.
Crystal Palace are outside the relegation zone in the Championship on goal difference.
Hart said: “I think it’s quite obvious I like a challenge, and to me, this is a big challenge.
“There is only one objective, and that is to stay in this division.
“I feel that maybe, with a few tweaks, we can get there.”