My Latin is rather rusty, but until someone more qualified to answer this question shows up, I guess I might as well take a stab at it.
I would say the normal way to negate that statement would be “Timor mortis non conturbat me.” However, in Latin you can play with the word order of a sentence without changing its meaning more easily than in English (which is a good thing when you’re trying to write in a certain meter), so perhaps moving the “non” wouldn’t change what you’re saying.
tim314 is correct. It should be “timor mortis non conturbat me.” In prose Latin, it would be very rare for a negation to be far away from its negatived verb. In poetry, though, it sometimes happens. As a side note, while Latin prose is certainly free with word order, probably the most likely ordering for the [original] sentence above would be with the verb last, “timor mortis me [non] conturbat.”