The question of stability of a planar solid-liquid interface in undercooled pure and alloy melts has been reconsidered without the restrictive assumption of no heat flow in the solid made in earlier works. The modified analysis indicates that provided the thermal gradient on the solid side of the interface, Gs, is positive, stability can be achieved in an undercooled alloy melt for growth rates R>Ra, whereas a recent analysis by Trivedi and Kurz, which assumes Gs = 0, suggests that stability is possible only if R>Ra + Rat. Here Ra is the familiar absolute stability limit of Mullins and Sekerka and Rat, is the absolute stability limit in an undercooled pure melt, as identified by Trivedi and Kurz. The absolute stability criterion for steady-state planar growth in an undercooled alloy melt is thus the same as derived earlier by Mullins and Sekerka for directional solidification. Relaxing the restrictive assumption of Gs = 0 also reveals that there is a regime of stability for low growth rates and low supercoolings. Stability is possible under these conditions if Gs>0, and the bath undercooling ΔTb < ΔTO + ΔTh/2, where ΔTO is the freezing range of the alloy and ΔTh is the hypercooling limit for the pure melt. For large supercoolings, Gs < 0, and the interface will be unstable with respect to large wavelength perturbations, even if R > Ra + Rat.