  We study a self-attractive random walk such that each trajectory of length
$N$ is penalised by a factor proportional to $\exp ( - |R_N|)$, where $R_N$ is
the set of sites visited by the walk. We show that the range of such a walk is
close to a solid Euclidean ball of radius approximately $\rho_d N^{1/(d+2)}$,
for some explicit constant $\rho_d >0$. This proves a conjecture of Bolthausen
who obtained this result in the case $d=2$.
