Diffusion least-mean-square (LMS) is an adaptive algorithm that estimates an unknown global vector from its linear measurements obtained at all nodes in a distributed manner when each node in the network needs to track the unknown vector in real-time. The algorithm uses the conventional average consensus protocol in order to combine neighbors' estimates at each node, while another protocol, consensus propagation (CP), is known to achieve faster and exact average consensus when the network has a tree structure. This paper proposes a novel diffusion LMS algorithm using CP, which can be applied for any network by extracting a spanning tree from the original network and can achieve the same solution as the centralized LMS in a fully distributed manner. This paper also proposes an algorithm by using the idea of loopy CP, so that it can be directly applied even when the network is not a tree and shows that its special case results in the diffusion LMS using a novel combination rule. Moreover, we optimize the constants involved in the proposed combination rule in terms of the steady-state mean-square-deviation of the diffusion LMS and show an adaptive implementation of the proposed algorithm. The simulation results demonstrate that the proposed algorithm using CP is beneficial for large-scale networks, and the diffusion LMS with the proposed combination rule achieves better convergence performance than that with the conventional combination rules when the measurement noise power depends on nodes.