  The parabolic Anderson problem is the Cauchy problem for the heat equation
$\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb
Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider
independent and identically distributed potential variables, such that
Prob$(\xi(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially
localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large
time $t$, the solution is completely localised in a single point with high
probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$
with values in $\Z^d$ such that $\lim_{t \uparrow\infty}
u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the
asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.
