  We introduce two notions of amenability for a Banach algebra $\cal A$. Let
$I$ be a closed two-sided ideal in $\cal A$, we say $\cal A$ is $I$-weakly
amenable if the first cohomology group of $\cal A$ with coefficients in the
dual space $I^*$ is zero; i.e., $H^1({\cal A},I^*)=\{0\}$, and, $\cal A$ is
ideally amenable if $\cal A$ is $I$-weakly amenable for every closed two-sided
ideal $I$ in $\cal A$. We relate these concepts to weak amenability of Banach
algebras. We also show that ideal amenability is different from amenability and
weak amenability. We study the $I$-weak amenability of a Banach algebra $\cal
A$ for some special closed two-sided\break ideal $I$.
