We study in detail locally Cohen-Macaulay curves in P4which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemesHd,g(2H) of lo-cally Cohen-Macaulay curves in 2Hof degreedand arithmetic genusg, and we show thatHd,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.