Phase-change materials (PCMs) are promising candidates for nonvolatile data storage and reconfigurable electronics, but high programming currents have presented a challenge to realize low-power operation. We controlled PCM bits with single-wall and small-diameter multi-wall carbon nanotubes. This configuration achieves programming currents of 0.5 microampere (set) and 5 microamperes (reset), two orders of magnitude lower than present state-of-the-art devices. Pulsed measurements enable memory switching with very low energy consumption. Analysis of over 100 devices finds that the programming voltage and energy are highly scalable and could be below 1 volt and single femtojoules per bit, respectively.