It is proved that an $L^1_{\varphi}$ space of analytic functions in the unit disc, with the weight $\varphi'(1-|z|)$, is isomorphic to the Lebesgue sequence space $l^1$ only if $\varphi$ is ``normal''. The converse is known from the papers of Shields and Williams [13] and Lindenstrauss and Pelczynski [4]. The key of our proof are three classical results: Paley's theorem on lacunary series, Pelczynski's theorem on complemented subspaces of $l^1$ and Lindenstrauss-Pelczynski's theorem on the equivalence of unconditional bases in $l^1$.