  The well-known 1-2-3 Conjecture asserts that the edges of every graph without
isolated edges can be weighted with $1$, $2$ and $3$ so that adjacent vertices
receive distinct weighted degrees. This is open in general. We prove that every
$d$-regular graph, $d\geq 2$, can be decomposed into at most $2$ subgraphs
(without isolated edges) fulfilling the 1-2-3 Conjecture if
$d\notin\{10,11,12,13,15,17\}$, and into at most $3$ such subgraphs in the
remaining cases. Additionally, we prove that in general every graph without
isolated edges can be decomposed into at most $24$ subgraphs fulfilling the
1-2-3 Conjecture, improving the previously best upper bound of $40$. Both
results are partly based on applications of the Lov\'asz Local Lemma.
