We have investigated electrical transport in a diffusive multiwalled carbon nanotube contacted using superconducting leads made of Al/Ti sandwich structure. We find proximity-induced superconductivity with measured critical currents up to Icm = 1.3 nA, tunable by gate voltage down to 10 pA. The supercurrent branch displays a finite zero bias resistance which varies as R0 ∝ I −α cm with α = 0.74. Using IV-characteristics of junctions with phase diffusion, a good agreement is obtained with 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)).