The organic radical-ion salt DEOCC-TCNQF4 contains linear chains of stacked molecules with significant Heisenberg antiferromagnet interactions along the chain and extremely weak interactions between the chains. Zero-field muSR has confirmed the absence of long-range magnetic order down to 20 mK and field-dependent muSR is found to be consistent with diffusive motion of the spin excitations. The anisotropic spin dynamics and the upper boundary for magnetic ordering temperature both indicate interchain magnetic coupling /J'/<7 mK. As the intrachain coupling J is 110 K, /J'/J/ is significantly less than 10(-4). This system therefore provides one of the most ideal examples of the one-dimensional S=1/2 Heisenberg antiferromagnet yet discovered.