  We present an investigation of the planar direct-current (dc)
paraconductivity of the model cuprate material HgBa$_2$CuO$_{4+\delta}$ in the
underdoped part of the phase diagram. The simple quadratic
temperature-dependence of the Fermi-liquid normal-state resistivity enables us
to extract the paraconductivity above the macroscopic $T_c$ with great
accuracy. The paraconductivity exhibits unusual exponential temperature
dependence, with a characteristic temperature scale that is distinct from
$T_c$. In the entire temperature range where it is discernable, the
paraconductivity is quantitatively explained by a simple superconducting
percolation model, which implies that underlying gap disorder dominates the
emergence of superconductivity.
