  In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the
minimum degree $k+1$ contains a cycle of length at least $\min\{n, 2(k+1)\}$.
Here we obtain a stability version of this result by characterizing those
graphs with minimum degree $k$ and circumference at most $2k+1$.
  We present applications of the above-stated result by obtaining generalized
Tur\'an numbers. In particular, for all $\ell \geq 5$ we determine how many
copies of a five-cycle as well as four-cycle are necessary to guarantee that
the graph has circumference larger than $\ell$. In addition, we give a new
proof of Luo's Theorem for cliques using our stability result.
