  The classical Banach space $L_1(L_p)$ consists of measurable scalar functions
$f$ on the unit square for which $$\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p
dy\Big)^{1/p}dx < \infty.$$ We show that $L_1(L_p)$ $(1 < p < \infty)$ is
primary, meaning that, whenever $L_1(L_p) = E\oplus F$ then either $E$ or $F$
is isomorphic to $L_1(L_p)$. More generally we show that $L_1(X)$ is primary,
for a large class of rearrangement invariant Banach function spaces.
