There exists a conformal diffeomorphism between $\mathbb{R}^3$ and $S_3$ (less a point): $$ g = dr^2+r^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right) $$ $$ r = R \tan \frac{\alpha}{2} $$

$$ g = \frac{R^2}{4\cos^4\frac{\alpha}{2}}\left[d\alpha^2+\sin^2\alpha\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)\right] $$

There also exists a conformal diffeomorphism between $\mathbb{R}^3$ (less a point) and the cylinder $\mathbb{R}\times S_2$:

$$ r = R\ e^{x/R} $$

$$ g=e^{2x/R}\left[dx^2+R^2\left(d\theta^2 + \sin^2\theta\ d\phi^2\right)\right] $$

There does not appear to be a conformal diffeomorphism between $\mathbb{R}^3$ (less a line) and the cylinder $\mathbb{R^2}\times S_1$. At least, my attempts to find one starting from $\mathbb{R^3}$ in cylindrical coordinates: $$ g = dz^2+ d\rho^2 +\rho^2 d\phi^2 $$ and remapping $(z, \rho)$ have failed. So my questions are:

1) Is it true that there is no such diffeomorphism?

2) If so, how do you show that? I think that local invariants like the Cotton tensor have nothing to say, because both $\mathbb{R}^3$ and $\mathbb{R^2}\times S_1$ are flat, and instead there is some kind of global obstruction.

Edited:

My original question was not sufficiently precise, it asked about existence of a conformal map, but I really meant existence of a conformal diffeomorphism. As Ben pointed out, there is the conformal map

$$ \mathbb{R}^3 \to \mathbb{R}^2 \times S_1 $$ $$ (x, y, z) \mapsto (x, y, \phi= \mathrm{mod}(z, 2\pi)) $$ but that is not injective and so not a diffeomorphism.