In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on Z penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by a weight exp(−h n |R n |), with |R n | the number of visited sites and h n a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at h n ≍ n 1/4 , inherent to the underlying lattice structure.