  The Erd\H{o}s-Gallai Theorem states that for $k \geq 2$, every graph of
average degree more than $k - 2$ contains a $k$-vertex path. This result is a
consequence of a stronger result of Kopylov: if $k$ is odd, $k=2t+1\geq 5$, $n
\geq (5t-3)/2$, and $G$ is an $n$-vertex $2$-connected graph with at least
$h(n,k,t) := {k-t \choose 2} + t(n -k+ t)$ edges, then $G$ contains a cycle of
length at least $k$ unless $G = H_{n,k,t} := K_n - E(K_{n - t})$.
  In this paper we prove a stability version of the Erd\H{o}s-Gallai Theorem:
we show that for all $n \geq 3t > 3$, and $k \in \{2t+1,2t + 2\}$, every
$n$-vertex 2-connected graph $G$ with $e(G) > h(n,k,t-1)$ either contains a
cycle of length at least $k$ or contains a set of $t$ vertices whose removal
gives a star forest. In particular, if $k = 2t + 1 \neq 7$, we show $G
\subseteq H_{n,k,t}$. The lower bound $e(G) > h(n,k,t-1)$ in these results is
tight and is smaller than Kopylov's bound $h(n,k,t)$ by a term of $n-t-O(1)$.
