We study the model-checking problem for WMTL, a cost-extension of the linear-time timed temporal logic MTL, that is interpreted over weighted timed automata. We draw a complete picture of the decidability for that problem: it is
decidable only for the class of one-clock weighted timed automata with a restricted stopwatch cost, and any slight extension
of this model leads to undecidability. We finally give some consequences on the undecidability of linear hybrid automata.