An experimental study to investigate the effect of an external magnetic field on the propagation of ion-acoustic waves (IAWs) has been carried out in hydrogen plasma containing two-temperature electrons and dust grains. It is a first step in understanding the propagation properties of IAWs in such an environment. A low-pressure hot cathode discharge method is chosen for plasma production. The desired two-electron groups with distinct temperatures are achieved by inserting two magnetic cages with a cusp-shaped magnetic field of different surface field strengths in the same chamber. The dust grains are dropped into the plasma with the help of a dust dropper, which gain negative charges by interacting with the plasma. The IAWs are excited with the help of a mesh grid inserted into the plasma. A planar Langmuir probe is used as a detector to detect the IAWs. The time-of-flight technique has been applied to measure the phase velocity of the IAWs. The results suggest that in the presence of a magnetic field, the phase velocity of IAWs increases, whereas introducing the dust particles leads to the lower phase velocity. The magnetic field is believed to have a significant effect on the wave damping. This study will aid in utilising IAWs as a diagnostic tool to estimate plasma parameters in the presence of an external magnetic field.