  An intersecting family of sets is trivial if all of its members share a
common element. Hilton and Milner proved a strong stability result for the
celebrated Erd\H{o}s--Ko--Rado theorem: when $n> 2k$, every non-trivial
intersecting family of $k$-subsets of $[n]$ has at most
$\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$ members. One extremal family
$\mathcal{HM}_{n, k}$ consists of a $k$-set $S$ and all $k$-subsets of $[n]$
containing a fixed element $x\not\in S$ and at least one element of $S$. We
prove a degree version of the Hilton--Milner theorem: if $n=\Omega(k^2)$ and
$\mathcal{F}$ is a non-trivial intersecting family of $k$-subsets of $[n]$,
then $\delta(\mathcal{F})\le \delta(\mathcal{HM}_{n.k})$, where
$\delta(\mathcal{F})$ denotes the minimum (vertex) degree of $\mathcal{F}$. Our
proof uses several fundamental results in extremal set theory, the concept of
kernels, and a new variant of the Erd\H{o}s--Ko--Rado theorem.
