  We show that if an exact special Lagrangian $N\subset \mathbb{C}^n$ has a
multiplicity one, cylindrical tangent cone of the form $\mathbb{R}^{k}\times
\mathbf{C}$ where $\mathbf{C}$ is a special Lagrangian cone with smooth,
connected link, then this tangent cone is unique provided $\mathbf{C}$
satisfies an integrability condition. This applies, for example, when
$\mathbf{C}= \mathbf{C}_{HL}^{m}$ is the Harvey-Lawson $T^{m-1}$ cone for $m\ne
8,9$.
