Suppose I have a smooth manifold with a vector bundle, and I have a connection.  If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-dimensional smooth manifold.)

I know that in general curvature-free connections do not exist, and that in general torsion-free connections do.  But is the existence of a curvature-free connection sufficient to prove the existence of a torsion-free connection?