Using pulsed NMR we have been studying the properties of the persistent precessing domain (PPD) in superfluid He-3-B at ultralow temperatures. These precessing spin domains are very long-lived, persisting for periods of over half an hour under favourable conditions. In the current experiment, by using two NMR coils, we are able to monitor the spatial location of the domain. The PPD is trapped in a field minimum along the axis of the cell, the position of which we can manipulate by variation of the field gradient. We find that the signal lifetime becomes much shorter and independent of temperature when the region of precession is brought into close proximity to the closed end of the experimental cell, suggesting that an extra temperature-independent dissipation mechanism comes into play in the vicinity of the cell walls.