  Let $(M,g,f)$ be a 3-dimensional complete steady gradient Ricci soliton.
Assume that $M$ is rectifiable, that is, the potential function can be written
as $f=f(r)$, where $r$ is a distance function. Then, we prove that $M$ is
isometric to (1) a quotient of $\mathbb{R}^3$, or (2) the Bryant soliton. In
particular, we show that any 3-dimensional complete rectifiable steady gradient
Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton.
Furthermore, we show that any $3$-dimensional complete rectifiable expanding
gradient Ricci soliton with positive Ricci curvature is rotationally symmetric.
