Yes, weapons can break while using elixirs, but the Immortal ability helps against this
From Ryoko's "Using Elixirs to Enhance Items" topic
Quote:
Steady: If the enhancement fails, it has better chances not to lose the previous enhancements.
Luck: Same as the lucky powder. Adds a little more luck on an item and improves your chance of success.
Imortality: Increases the chances your weapon won't break.