Very low frequency (VLF) radio signals with travel times~100 ms were observed continuously for up to~11 hr at night on Rarotonga (Cook Islands,~21°S) at 21.4 kHz from U.S. Navy transmitter NPM, Hawaii (~21°N). These signals travelled in the whistler-mode on well-defined paths, though not field-aligned ducts, through the ionospheric F region, and across the equator reaching altitudes~700-1,400 km depending on time of night. These same signals were also observed simultaneously in Dunedin (46°S), New Zealand, with very nearly the same travel times but with somewhat lower amplitudes and occurrence rates, consistent with the whistler-mode part of the propagation being at very low latitudes. Both sets of signals had similar Doppler shifts, typically tens of mHz, but sometimes up to a few hundred mHz, being positive during most of the night, while the whistler-mode group delays decreased due to both the shortening of the path and the decay of the near equatorial ionosphere, but negative near dawn when the Sun's rays start ionizing the F region. The signals are not observable during the day, fading out during dawn, due to increasing attenuation from the increasing electron density, and hence increasing collisions, in both the D and F regions. Similar weaker NPM signals were also seen at Rothera (68°S). In addition, similar 24.8-kHz signals were seen from the more distant NLK (Seattle,~48°N) at Rarotonga, though clearly weaker than from NPM, but not at Dunedin.