  In this paper we consider the heat equation with memory in a bounded region
$\Omega \subset\mathbb{R}^d$, $d\geq 1$, in the case that the propagation speed
of the signal is infinite (i.e. the Colemann-Gurtin model). The memory kernel
is of class $C^1$. We examine its controllability properties both under the
action of boundary controls or when the controls are distributed in a subregion
of $\Omega$. We prove approximate controllability of the system and, in
contrast with this, we prove the existence of initial conditions which cannot
be steered to hit the target $0$ in a certain time $T$, of course when the
memory kernel is not identically zero. In both the cases we derive our results
from well known properties of the heat equation.
