An associative equilibrium theory describing the sharp melting behavior of polymer-DNA hybrids is developed. The theory considers linear polymers with attached DNAs on each polymer that serve as "stickers" and with a two-state model governing the DNA melting equilibrium. For three or more oligonucleotides on each polymer, solutions of polymer-DNA hybrids are found to undergo phase separation at sufficiently low temperatures. The dense phase dissolves as temperature increases, which leads to a sharp increase in the fraction of non-hybridized DNA near the phase transition temperature, in agreement with experimental absorbance profiles at 260 nm. The melting temperature is predicted to have the same dependence on salt concentration as a solution of unattached DNAs and be weakly sensitive to the concentration of DNA in solution. The melting temperature is predicted to be higher than that of unattached DNA in solution, with the magnitude of the increase sensitive to the DNA hybridization cooperativity. The theoretical predictions are generally in good quantitative agreement with new experimental data (also presented here), which show the effect of the polymer-DNA hybrid length and salt concentration on the melting profiles.