  For an increasing sequence $(\omega_n)$ of algebra weights on $\mathbb R^+$
we study various properties of the Fr\'{e}chet algebra $A(\omega)=\bigcap_n
L^1(\omega_n)$ obtained as the intersection of the weighted Banach algebras
$L^1(\omega_n)$. We show that every endomorphism of $A(\omega)$ is standard, if
for all n\in\mathbb N$ there exists $m\in\mathbb N$ such that
$\omega_m(t)/\omega_n(t)\to\infty$ as $t\to\infty$. Moreover, we characterise
the continuous derivations on this algebra: If for all $n\in\mathbb N$ there
exists $m\in\mathbb N$ such that $t*\omega_n(t)/\omega_m(t)$ is bounded on
$\mathbb R^+$, then the continuous derivations on $A(\omega)$ are exactly the
linear maps $D$ of the form $D(f)=(Xf)*\mu$ for $f\in A(\omega)$, where $\mu$
is a measure in $B(\omega)=\bigcap_n M(\omega_n)$ and $(Xf)(t)=tf(t)$ for
$t\in\mathbb R^+$ and $f\in A(\omega)$. If the condition is not satisfied, we
show that $A(\omega)$ has no non-zero derivations.
