  We consider a one-dimensional random walk among biased i.i.d. conductances,
in the case where the random walk is transient but sub-ballistic: this occurs
when the conductances have a heavy-tail at $+\infty$ or at $0$. We prove that
the scaling limit of the process is the inverse of an $\alpha$-stable
subordinator, which indicates an aging phenomenon, expressed in terms of the
generalized arcsine law. In analogy with the case of an i.i.d. random
environment studied in details in [Enriquez, Sabot, Zindy, Bull. Soc. Math.
2009; Enriquez, Sabot, Tournier, Zindy, Ann. Appl. Probab. 2013], some `traps'
are responsible for the slowdown of the random walk. However, the phenomenology
is somehow different (and richer) here. In particular, three types of traps may
occur, depending on the fine properties of the tails of the conductances: (i) a
very large conductance (a well in the potential); (ii) a very small conductance
(a wall in the potential); (iii) the combination of a large conductance
followed shortly after by a small conductance (a well-and-wall in the
potential).
