Interaction of a field-driven magnetic domain wall with a correlated (line) defect is examined by Kerr imaging in subnanometer thin Co films. The line defect directs and confines the wall near the bottom of the effective potential trough U(eff), which competes with underlying random disorder that roughens the wall. We observe a kinetic "deroughening" with roughness exponent zeta approximately 0.1 well below zeta = 2/3 characteristic of random defects. Deroughening occurs on lengths greater than an inherent elastic screening length L(el), which is consistently explained by the restoring action of U(eff).