On a finite graph, there is a natural family of Boltzmann probability measures on cycle-rooted spanning forests, parametrized by weights on cycles. For a certain subclass of those weights, we construct Gibbs measures in infinite volume, as limits of probability measures on cycle-rooted spanning forests of increasing sequences of finite graphs. Those probability measures extend the family of already known random spanning forests and can be sampled by a random walks algorithm which generalizes Wilson's algorithm. We show that, unlike for uniform spanning forests, almost surely, all connected components are finite and two-points correlations decrease exponentially fast with the distance.