  We prove: (i) if $G$ is a 1-tough graph of order $n$ and minimum degree
$\delta$ with $\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating
cycle unless $G$ belongs to an easily specified class of graphs with
$\kappa(G)=2$ and $\tau(G)=1$. The second result follows immediately from the
first result: (ii) if $G$ is a 3-connected 1-tough graph with
$\delta\ge(n-2)/3$ then each longest cycle in $G$ is a dominating cycle.
