Let A be a locally $m$-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet
$A-$module $X=A+/I$ to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we
introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that $X$ is strictly
flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a
locally $m$-convex Fréchet algebra $A$ is amenable if and only if $A$ is isomorphic to a reduced inverse limit of
amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and
by Helemskii and Sheinberg to the setting of locally $m$-convex Fréchet algebras. As a corollary, we show that
Connes and Haagerup's theorem on amenable $C*$-algebras and Sheinberg's theorem on amenable uniform
algebras hold in the Fréchet algebra case. We also show that a quasinormable locally $m$-convex Fréchet algebra
has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally
$m$-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i.