  We prove a "splicing formula" for the LMO invariant, which is the universal
finite-type invariant of rational homology $3$-spheres. Specifically, if a
rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two
framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology
$3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the
Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the
techniques that Bar-Natan and Lawrence developed to obtain a rational surgery
formula for the LMO invariant. In low degrees, we recover Fujita's formula for
the Casson-Walker invariant and we observe that the second term of the Ohtsuki
series is not additive under "standard" splicing. The splicing formula also
works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$,
hence we get a "satellite formula" for the Kontsevich-LMO invariant.
