An experimental investigation was performed on self-sustained, axis-symmetric, laminar diffusion flames in which the fuel, heptane, was injected as a monodisperse spray in a co-flow of air. The appearance of such flames was in all respects similar to that of a candle. The inner core of the flame was a relatively dark region, in which the droplets simply evaporated at relatively low temperatures, but did not burn. This inner core was enveloped by a higher temperature region where soot was present. The flame was located immediately outside of the sooty region and was characterized by a soft, deep blue luminescence. The droplet life-history along the centerline of three flames was examined by phase Doppler anemometry and sizing, as well as thermocouple measurements. It was found that the D-square law with constant evaporation coefficient was satisfied over a significant portion of the droplet lifetimes. The evaporation coefficients extracted from such measurements range from 0.53 to 0.60 mm2/s, values that are on average 35 % to 45% lower than those calculated for an isolated droplet in a convective atmosphere of N2. It was also shown that an average evaporation coefficient for the droplets in the flame could be determined by simply measuring the height of the dark, vaporization region and the liquid flow rate, without the need for in-situ measurements. No evidence of individual droplet burning was found. Rather, conditions were more akin to what is often labeled sheath combustion, with a continuous flame enveloping the entire droplet region and a relatively cooler core in which droplets are evaporating but not burning. Droplet-droplet interactions, although certainly present near the flame surface, do not seem to be significant enough in the flame core to cause a departure from the D-square law.