  We report low-temperature specific heat, $C(T)$, measurements on
(Yb$_{1-x}$Lu$_x$)$_4$As$_3$ with $x=0.01$ and $x=0.03$, where nonmagnetic Lu
atoms are randomly distributed on antiferromagnetic $S=1/2$ Heisenberg chains
with $J/k_{\mathrm B}=28$ K. The observed reduction of $C$ below 15 K with
increasing $x$ is accurately described by quantum transfer matrix simulations
without any adjustable parameter, implying that the system is an excellent
experimental realization of segmented quantum spin chains. Finite-size effects
consistent with conformal-field theory predictions are leading to the formation
of an effective low-energy gap. The size of the gap increases with Lu content
and accounts for the impurity driven reduction of the specific heat. For both
concentrations our results verify experimentally the low temperature scaling
behavior established theoretically and also confirm the value of $J$ determined
from pure Yb$_4$As$_3$.
