The electron transport properties of the thermodynamically stable Al₆₈Cu₁₇Ru₁₅ quasicrystal have been studied through the measurements of the electronic specific heat coefficient and the temperature dependence of the electrical resistivity in the range 4.2-300 K. The full-width at the half maximum for the strongest X-ray diffraction line (100000) is reduced to be less than 0.15 nm^-1 either by remelting the ingot with subsequent furnace cooling or by annealing the melt-spun ribbon at 850 degrees C for 24 hours. An apparent improvement in the quasicrystallinity upon the heat-treatment caused a drastic increase in resistivity up to 1600 mu Omega cm and accompanied a very small electronic specific heat coefficient gamma lower than 0.3 mJ mol^-1 K^-2. The temperature dependence of the resistivity characterised by a concave curvature with a negative TCR can be discussed in terms of the weak localisation of conduction electrons. The authors are also convinced that the thermodynamically stable quasicrystals like the present Al-Cu-Ru and the previously studied Al-Li-Cu always exhibit a very low gamma value coupled with a high resistivity. From this they conclude that the electron density of states in thermodynamically stable quasicrystals possesses a structure-induced minimum and that the Fermi level does fall in this critical range. An increase in the resistivity upon improvement in quasicrystallinity has been discussed in terms of the generalised Faber-Ziman theory and also in terms of a possible enhancement in the coherent multiple scattering due to locally well-developed short-range order.