  Let $M_1$ and $M_2$ be special Lagrangian submanifolds of a compact
Calabi-Yau manifold $X$ that intersect transversely at a single point. We can
then think of $M_1\cup M_2$ as a singular special Lagrangian submanifold of $X$
with a single isolated singularity. We investigate when we can regularize
$M_1\cup M_2$ in the following sense: There exists a family of Calabi-Yau
structures $X_\alpha$ on $X$ and a family of special Lagrangian submanifolds
$M_\alpha$ of $X_\alpha$ such that $M_\alpha$ converges to $M_1\cup M_2$ and
$X_\alpha$ converges to the original Calabi-Yau structure on $X$. We prove that
a regularization exists in two key cases: (1) when the complex dimension of $X$
is three, $\Hol(X)=\SU(3)$, and $[M_1]$ is not a multiple of $[M_2]$ in
$H_3(X)$, and (2) when $X$ is a torus with complex dimension at least three,
$M_1$ is flat, and the intersection of $M_1$ and $M_2$ satisfies a certain
angle criterion. One can easily construct examples of the second case, and thus
as a corollary we construct new examples of non-flat special Lagrangian
submanifolds of Calabi-Yau tori.
