Let $S$ be a Noetherian scheme, let $Y$ be a scheme of finite type over $S$, and let $X$ be an algebraic space of finite type over $S$. Suppose that there is a morphism $f:Y \rightarrow X$ which is proper and birational. Must $X$ be a scheme?


See also 
[When is an algebraic space a scheme?](https://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme)



Algebraic space birational to a scheme
I am just posting my comment as an answer. Chow's Lemma for algebraic spaces is Theorem IV.3.1, p. 192 of the following.


MR0302647 (46 #1791)   

Knutson, Donald   

Algebraic spaces.   

Lecture Notes in Mathematics, Vol. 203.   

Springer-Verlag, Berlin-New York, 1971. vi+261 pp.   



For every separated, Noetherian scheme $S$, for every finitely presented, separated morphism of algebraic spaces, $X\to S$, there exists a morphism $f:Y\to X$ that is projective and birational such that $Y\to S$ is a quasi-projective morphism.


