We have investigated electrical transport in a diffusive multiwalled carbon nanotube contacted using superconducting leads made of an Al/Ti sandwich structure. We find proximity-induced superconductivity with measured critical currents up to I(cm)=1.3 nA, tunable by the gate voltage down to 10 pA. The supercurrent branch displays a finite zero bias resistance which varies as R(0) proportional to I(cm){-alpha} with alpha=0.74. Using IV characteristics of junctions with phase diffusion, a good agreement is obtained with the Josephson coupling energy in the long, diffusive junction model of A. D. Zaikin and G. F. Zharkov [Sov. J. Low Temp. Phys. 7, 184 (1981)].