We experimentally study the excitation and propagation of acoustic solitary waves in a one-dimensional dusty plasma (i.e. a Yukawa chain) with $n=65$ particles interacting through a screened Coulomb potential. The lattice constant $a=1.02\pm 0.02$ mm. Waves are launched by applying a 100 mW laser pulse to one end of the chain for laser pulse durations from 0.10 to 2.0 s. We observe damped solitary waves which propagate for distances ${\gtrsim}30a$ with an acoustic speed $c_{s}=11.5\pm 0.2~\text{mm}~\text{s}^{-1}$. The maximum velocity perturbation increases with laser pulse duration for durations ${\leqslant}0.5$ s and then saturates at ${\approx}15\,\%$. The wave speed is found to be independent of the maximum amplitude, indicating that the formation of nonlinear solitons is prevented by neutral-gas damping.