Good question. Depends on what “conviction” means.
Generally a conviction destroys the civil case.
Here however, if Zimmerman gets convicted because the jury pool was tainted by the false MSM coverage, then his damages go through the roof. (And more important, it would be great grounds for appeal and eventual reversal of the conviction). So ultimately there wouldn’t be a conviction in the criminal case, but a multi-billion dollar award against NBC, ABC, et al, ad nauseum, in the civil case.