Efficient transport of spin angular momentum is expected to play a crucial role in future spintronic devices, which will potentially operate at frequencies reaching the terahertz range. On the other hand, antiferromagnetic insulators exhibit significant potential for facilitating ultrafast pure spin currents by terahertz magnons. Consequently, we here use femtosecond laser pulses to trigger ultrafast spin currents across thin films of the model antiferromagnet NiO in Py|NiO|Pt stacks, where permalloy (Py) and Pt serve as spin-current source and detector, respectively. We find that the spin-current pulses traversing NiO reach a velocity up to 40 nm/ps and experience increasing attenuation, delay, and broadening as the NiO thickness is increased. We can consistently explain our observations by ballistic transport of incoherent magnons. The relaxation length of 20 nm is 1 order of magnitude larger than for ballistic conduction-electron spin transport in metals and, thus, relevant for potential applications. Methodologically, our all-optical and contact-free spin-conductance approach has large potential to characterize terahertz magnon transport in insulators with any kind of magnetic order.