 Solution

For every infinite $\alpha <\kappa$, the generic union at coordinate $\alpha$ gives a function $g_{\alpha }:\omega \to \alpha$. The requirement to assign $(\alpha ,n)$ is dense for each $n$, and the requirement to use any specified value $\beta <\alpha$ is dense by assigning it at a fresh natural-number position. Therefore $g_{\alpha }$ is a surjection, and every ordinal below $\kappa$ becomes countable.

The $\kappa$-chain condition preserves the regularity of $\kappa$. For a direct verification, a forcing name for a function from some $\mu <\kappa$ into $\kappa$ has fewer than $\kappa$ possible values at each coordinate, using a maximal deciding forcing antichain. The union of these possible-value sets has size less than $\kappa$ by regularity and is bounded in $\kappa$. No such function can be cofinal. In particular $\kappa$ remains uncountable, while every smaller ordinal is countable. HenceThis is the finite Lévy collapse to omega-one.