Experimental results are presented on combustion of freely-falling droplets in an apparatus that promotes spherical symmetry. Design of the apparatus is based upon the principle that gravitationally-induced pressure gradients in gases may be significantly reduced by accelerating the gases with properly-contoured tubes. Droplets travelling with the gases experience substantially reduced forced-convective and buoyant (gravitational) flow effects, strongly promoting spherically symmetrical combustion. It was found that burning rates for single-component alkane fuel droplets were significantly lower in our contoured tube than in a straight tube where similar initial conditions were applied. The results suggest that use of a contoured drop-tube significantly diminished the degree of natural and forced convection experienced by the falling droplets, allowing droplet burning rate measurements to be made under very weak buoyant and forced-convective flow conditions. This is desirable so that fundamental results can be obtained that can be compared to simplified models that neglect buoyant and forced convection.