Contrast and quality of ultrasound images are adversely affected by the excessive presence of speckle. However, being an inherent imaging property, speckle helps in tissue characterization and tracking. Thus, despeckling of the ultrasound images requires the reduction of speckle extent without any oversmoothing. In this letter, we aim to address the despeckling problem using an unsupervised deep adversarial approach. A despeckling residual neural network (DRNN) is trained with an adversarial loss imposed by a discriminator. The discriminator tries to differentiate between the despeckled images generated by the DRNN and the set of high-quality images. Further to prevent the developed DRNN from oversmoothing, a structural loss term is used along with the adversarial loss. Experimental evaluations show that the proposed DRNN is able to outperform the state-of-the-art despeckling approaches.