  In this paper a bijection between the set of prime ideals of a Leavitt path
algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the
prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient
conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is
primitive are also found.
