We report a new scheme for fabrication of clean, suspended superconducting weak links from pristine single-walled carbon nanotubes (SWCNT). The SWCNTs were grown using the floating-catalyst chemical vapour deposition (FC-CVD) and directly deposited on top of prefabricated superconducting molybdenum-rhenium (MoRe) electrodes by thermophoresis at nearly ambient conditions. Transparent contacts to SWCNTs were obtained by vacuum-annealing the devices at 900 °C, which enabled proximity-induced supercurrents up to 53 nA. SWCNT weak links fabricated on MoRe/palladium bilayer sustained supercurrents up to 0.4 nA after annealing at relatively low temperature of 220 °C. The fabrication process does neither expose SWCNTs to lithographic chemicals, nor the contact electrodes to the harsh conditions of in situ CVD growth. Our scheme facilitates new experimental possibilities for hybrid superconducting devices.