Ultrasonic welding (USW) is one of the promising method to join thin metals and wires, providing strength and electric conductivity, in addition to avoid metallurgical defects, such as the formation of intermetallic compounds, brittle phases, and porosities. However, USW is a very sensitive process, which depends on many parameters. In the present work, the influence of ultrasonic welding parameters such as weld time and pressure on the joint strength of copper wires of diameters 1.6 and 1.8 mm is analyzed. A suitable experimental design based on Taguchi’s robust design methodology was designed and executed for conducting experiments. The analysis of variance (ANOVA) and signal-to-noise ratio analyses are employed to investigate the influence of different welding parameters on the weld strength and to obtain the optimum parameters. The maximum weld strength obtained for copper wires of diameters 1.6 and 1.8 mm are 7.45 MPa and 10.48 MPa, respectively. The analysis of variance (ANOVA) study explored that diameter, pressure, and interaction between diameter and pressure, diameter and weld time have the most impact on the weld strength.