An identifying code in a graph $G$ is a subset of vertices with the property that for each vertex $v \in V(G)$, the collection of elements of $C$ at distance at most $1$ from $v$ is non-empty and distinct from the collection of any other vertex. We consider the minimum density $d^*({\cal S}_k)$ of an identifying code in the square grid ${\cal S}_k$ of height $k$ (i.e. with vertex set $ \mathbb{Z} \times \{1, \dots , k\}$). Using the Discharging Method, we prove $\displaystyle \frac{7}{20} + \frac{1}{20k} \leq d^*({\cal S}_k) \leq \min \left\{\frac{2}{5}, \frac{7}{20} + \frac{3}{10k} \right\}$, and $\displaystyle d^*({\cal S}_3) =\frac{3}{7}$.