  The set $\mathit{RT}(M)$ of values of the
$\mathit{SL}(2,\mathbb{C})$-Reidemeister torsion of a 3-manifold $M$ can be
both finite and infinite. We prove that $\mathit{RT}(M)$ is a finite set if $M$
is the splice of two certain knots in the 3-sphere. The proof is based on an
observation on the character varieties and $A$-polynomials of knots.
