We study the impact of a weak bond disorder on the spinon heat transport in the S=1/2 antiferromagnetic (AFM) Heisenberg chain material Sr _1-xCa _xCuO ₂. We observe a drastic suppression in the magnetic heat conductivity κ _mag even at tiny disorder levels (i.e., Ca-doping levels), in stark contrast to previous findings for κ _mag of S=1/2 two-dimensional square lattice and two-leg spin-ladder systems, where a similar bond disorder has no effect on κ _mag. Hence, our results underpin the exceptional role of integrability of the S=1/2 AFM Heisenberg chain model and suggest that the bond disorder effectively destroys the ballistic nature of its heat transport. We further show that the suppression of κ _mag is captured by an effective spinon-impurity scattering length, which exhibits the same doping dependence as the long-distance exponential decay length of the spin-spin correlation as determined by density-matrix renormalization group calculations.