  We study the extensions of two left modules $W_1, W_2$ for a meromorphic
open-string vertex algebra $V$. We show that the extensions satisfying some
technical but natural convergence conditions are in bijective correspondence to
the first cohomology classes associated to the $V$-bimodule $\mathcal{H}_N(W_1,
W_2)$ constructed in \cite{HQ-Red}. When $V$ is grading-restricted and contains
a nice vertex subalgebra $V_0$, those convergence conditions hold
automatically. In addition, we show that the dimension of $\text{Ext}^1(W_1,
W_2)$ is bounded above by the fusion rule $N\binom{W_2}{VW_1}$ in the category
of $V_0$-modules. In particular, if the fusion rule is finite, then
$\text{Ext}^1(W_1, W_2)$ is finite-dimensional. We also give an example of an
abelian category consisting of certain modules of the Virasoro VOA that does
not contain any nice subalgebras, while the convergence conditions hold for
every object.
