We shall say that a solvable Lie algebra L L is a minimal nonnilpotent Lie algebra if L L is nonnilpotent but all proper subalgebras of L L are nilpotent. It is shown here that if L L is a minimal nonnilpotent Lie algebra, then L L is the vector space direct sum of A A and F F where A A is an ideal in L , F L,F is a one-dimensional subalgebra of L L , either A A is a minimal ideal of L L or the center of A A coincides with the derived algebra, A ′ A’ , of A A and in either case F F acts irreducibly on A / A ′ A/A’ .