  We prove that, when elliptic curves $E/\mathbb{Q}$ are ordered by height, the
average number of integral points $\#|E(\mathbb{Z})|$ is bounded, and in fact
is less than $66$ (and at most $\frac{8}{9}$ on the minimalist conjecture). By
"$E(\mathbb{Z})$" we mean the integral points on the corresponding quasiminimal
Weierstrass model $E_{A,B}: y^2 = x^3 + Ax + B$ with which one computes the
na\"{\i}ve height. The methods combine ideas from work of Silverman, Helfgott,
and Helfgott-Venkatesh with work of Bhargava-Shankar and a careful analysis of
local heights for "most" elliptic curves. The same methods work to bound
integral points on average over the families $y^2 = x^3 + B$, $y^2 = x^3 + Ax$,
and $y^2 = x^3 - D^2 x$.
