  We present a stable uniqueness theorem for non-unital C*-algebras.
Generalized tracial rank one is defined for stably projectionless simple
C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple
amenable C*-algebras with $gTR(A)\le 1$and $gTR(B)\le 1.$ Suppose also that
$KK(A, D)=KK(B,D)=\{0\}$ for all C*-algebras $D.$ Then $A\cong B$ if and only
if they have the same tracial cones with scales. We also show that every
separable simple C*-algebra, $A$ with finite nuclear dimension which satisfies
the UCT with non-zero traces must have $gTR(A)\le 1$ if $K_0(A)$ is torsion. In
the next part of this research, we show similar results without the restriction
on $K$-theory.
