  Near the melting transition the bending elastic constant, $\kappa$, an
emergent property of double-stranded DNA (dsDNA), is shown not to follow the
rod-like scaling for small length $N$. The reduction in $\kappa$ with
temperature is determined by the denatured bubbles for a continuous transition,
e.g., when the two strands are gaussian, but by the broken bonds near the open
end in a Y-like configuration for a first-order transition as for strands with
excluded volume interactions. In the latter case, a lever rule is operational
implying a phase coexistence although dsDNA is known to be a single phase.
