Heterostructure core-shell semiconductor nanowires (NWs) have attracted tremendous interest recently due to their remarkable properties and potential applications as building blocks for nanodevices. Among their unique traits, thermal properties would play a significant role in thermal management of future heterostructure NW-based nanoelectronics, nanophotonics, and energy conversion devices, yet have been explored much less than others. Similar to their electronic counterparts, phonon spectrum and thermal transport properties could be modified by confinement effects and the acoustic mismatch at the core-shell interface in small diameter NWs (<20 nm). However, fundamental thermal measurement on thin core shell NWs has been challenging due to their small size and their expected low thermal conductivity (κ). Herein, we have developed an experimental technique with drastically improved sensitivity capable of measuring thermal conductance values down to ∼10 pW/K. Thermal conductivities of Ge and Ge-Si core-shell NWs with diameters less than 20 nm have been measured. Comparing the experimental data with Boltzmann transport models reveals that thermal conductivities of the sub-20 nm diameter NWs are further suppressed by the phonon confinement effect beyond the diffusive boundary scattering limit. Interestingly, core-shell NWs exhibit different temperature dependence in κ and show a lower κ from 300 to 388 K compared to Ge NWs, indicating the important effect of the core-shell interface on phonon transport, consistent with recent molecular dynamics studies. Our results could open up applications of Ge-Si core shell NWs for nanostructured thermoelectrics, as well as a new realm of tuning thermal conductivity by "phononic engineering".