Simulating uniformly on $S^1=\{x \in \mathbb{R}^n \mid \|x\|_1=1\}$

Flip $n$ fair coins to pick an orthant—that is, to pick the signs of the coordinates of the point you are choosing. Now pick a point uniformly in the standard simplex, and flip the signs of its coordinates according to what the coins told you.

Does the following work? (read the comment of joriki to convince yourself that the algorithm works --- thanks joriki)

Cut a line of length 1 into $n$ parts. Mathematically, throw $n-1$ random numbers between uniform between 0 and 1. Sort them to obtain $\mathbf{s}=(0,s_1, \dots, s_{n-1},1)$ with $s_i \leq s_j$; $\forall i <j$.

Take the point $\mathbf{x}= \left[\sigma_1(s_1- s_0), \sigma_2(s_2 - s_1), \dots, \sigma_n(s_n - s_{n-1})\right] \in S^1$. Here, $\sigma_i = \pm 1$ are randomly choosen. It is quite clear that $\mathbf{x}$ is on the unit sphere. The question remains: is it uniform on the sphere?