  For the first time, we observed and controlled the Brownian motion of
solitons. We launched solitonic excitations in highly elongated $^{87}\rm{Rb}$
BECs and showed that a dilute background of impurity atoms in a different
internal state dramatically affects the soliton. With no impurities and in
one-dimension (1-D), these solitons would have an infinite lifetime, a
consequence of integrability. In our experiment, the added impurities scatter
off the much larger soliton, contributing to its Brownian motion and decreasing
its lifetime. We describe the soliton's diffusive behavior using a quasi-1-D
scattering theory of impurity atoms interacting with a soliton, giving
diffusion coefficients consistent with experiment.
