  The purpose of this paper is to study a one-dimensional polymer penalized by
its range and placed in a random environment $\omega$. The law of the simple
symmetric random walk up to time $n$ is modified by the exponential of the sum
of $\beta \omega_z - h$ sitting on its range, with~$h$ and $\beta$ positive
parameters. It is known that, at first order, the polymer folds itself to a
segment of optimal size $c_h n^{1/3}$ with $c_h = \pi^{2/3} h^{-1/3}$. Here we
study how disorder influences finer quantities. If the random variables
$\omega_z$ are i.i.d.\ with a finite second moment, we prove that the left-most
point of the range is located near $-u_* n^{1/3}$, where $u_* \in [0,c_h]$ is a
constant that only depends on the disorder. This contrast with the homogeneous
model (i.e. when $\beta=0$), where the left-most point has a random location
between $-c_h n^{1/3}$ and $0$. With an additional moment assumption, we are
able to show that the left-most point of the range is at distance $\mathcal U
n^{2/9}$ from $-u_* n^{1/3}$ and the right-most point at distance $\mathcal V
n^{2/9}$ from $(c_h-u_*) n^{1/3}$. Here again, $\mathcal{U}$ and $\mathcal{V}$
are constants that depend only on $\omega$.

Probability