Fusion neutrons from a heavy water droplet target irradiated with laser pulses of 3 x 10(19) W/cm(2) and from a deuterated secondary target are observed by a time-of-flight (TOF) neutron spectrometer. The observed TOF spectrum can be explained by fusion of deuterium ions simultaneously originating from two different sources: ion acceleration in the laser focus by ponderomotively induced charge separation and target-normal sheath acceleration off the target rear surface. The experimental findings agree well with 3D particle-in-cell simulations.