  Suppose that $\Omega$ is a lattice in the complex plane and let $\sigma$ be
the corresponding Weierstrass $\sigma$-function. Assume that the point $\tau$
associated to $\Omega$ in the standard fundamental domain has imaginary part at
most 1.9. Assuming that $\Omega$ has algebraic invariants $g_2,g_3$ we show
that a bound of the form $c d^m (\log H)^n$ holds for the number of algebraic
points of height at most $H$ and degree at most $d$ lying on the graph of
$\sigma$. To prove this we apply results by Masser and Besson. What is perhaps
surprising is that we are able to establish such a bound for the whole graph,
rather than some restriction. We prove a similar result when, instead of
$g_2,g_3$, the lattice points are algebraic. For this we naturally exclude
those $(z,\sigma(z))$ for which $z\in\Omega$.
