We study the impact of a weak bond disorder on the spinon heat transport in the antiferromagnetic (AFM) Heisenberg chain material . We observe a drastic suppression in the magnetic heat conductivity even at tiny disorder levels (i.e., Ca-doping levels), in stark contrast to previous findings for of two-dimensional square lattice and two-leg spin-ladder systems, where a similar bond disorder has no effect on . Hence, our results underpin the exceptional role of integrability of the 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 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.