It is known that not every Banach algebra has non-trivial bounded derivations. For instance, consider large families of weighted semisimple Banach algebras. In particular, we will be concerned with derivations within the concrete frame of the non-abelian, non-unitary, involutive Banach algebra $l^{2}(N^{2})$. The theoretical interest in this algebra is based on the well-known fact that it is isomorphic to the class of Hilbert-Schmidt operators acting between two given separable Hilbert spaces. In this article, we characterize and determine the explicit structure of all bounded derivations on $l^{2}(N^{2})$.