  Let $G=(V, E)$ be a given edge-weighted graph and let its {\em realization}
$\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e \in E$
independently with probability $p$. In the {\em stochastic matching} problem,
the goal is to pick a sparse subgraph $Q$ of $G$ without knowing the
realization $\mathcal{G}$, such that the maximum weight matching among the
realized edges of $Q$ (i.e. graph $Q \cap \mathcal{G}$) in expectation
approximates the maximum weight matching of the whole realization
$\mathcal{G}$.
  In this paper, we prove that for any desirably small $\epsilon \in (0, 1)$,
every graph $G$ has a subgraph $Q$ that guarantees a
$(1-\epsilon)$-approximation and has maximum degree only $O_{\epsilon, p}(1)$.
That is, the maximum degree of $Q$ depends only on $\epsilon$ and $p$ (both of
which are known to be necessary) and not for example on the number of nodes in
$G$, the edge-weights, etc.
  The stochastic matching problem has been studied extensively on both weighted
and unweighted graphs. Previously, only existence of (close to)
half-approximate subgraphs was known for weighted graphs [Yamaguchi and
Maehara, SODA'18; Behnezhad et al., SODA'19]. Our result substantially improves
over these works, matches the state-of-the-art for unweighted graphs [Behnezhad
et al., STOC'20], and essentially settles the approximation factor.
