Let $X$ be a smooth finite type separated connected Deligne-Mumford stack over $\mathbb C$.


Does there exist a finite etale morphism $Y\to X$ with $Y$ a scheme?


What if $X$ is an algebraic space (i.e., trivial stabilizers)?


Edit: I changed the old question to a different question which should be more clear. An answer to the new question would help a lot in answering the old question.



Finite etale atlas for Deligne-Mumford stacks
It seems to me that the answer is NO if $X$ is a DM stack. If I'm not mistaken, it suffices to give a smooth finite type separated connected Deligne-Mumford stack over $\mathbb{C}$ which is simply connected (since such a thing has no non-trivial finite etale covers, let alone finite etale covers by a scheme). But [this paper](http://arxiv.org/pdf/math/0504309.pdf) of Behrend and Noohi shows that the weighted projective lines $\mathcal{P}(m, n)$ (constructed by taking the stack quotient of $\mathbb{A}^2\setminus\{0\}$ by the $\mathbb{G}\_m$-action $\lambda\cdot(x,y):=(\lambda^m x, \lambda^n y)$) are simply connected. The proof is easy; one just uses the long exact sequence for homotopy groups associated to the fibration $$\mathbb{G}\_m\to \mathbb{A}^2\setminus\{0\}\to \mathcal{P}(m, n).$$


**Added later**: The answer seems to be no for algebraic spaces as well. Example 5.7 [here](http://projecteuclid.org/download/pdf_1/euclid.pja/1117805146) is simply connected if I'm not mistaken, and is not a scheme by Remark 3.4 in the same paper.


