We give a new construction of Ramanujan graphs using a generalized type of covering graph called a weighted covering graph. For a given prime p the basic construction produces bipartite Ramanujan graphs with $4p$ vertices and degrees 2N where roughly $p + 1 - \sqrt{2p} < N \le p$. We then give generalizations to produce Ramanujan graphs of other sizes and degrees as well as general results about base graphs which have weighted covers that satisfy their Ramanujan bounds. To do the construction, we define weighted covering graphs and distinguish a subclass of Galois weighted covers that allows for block diagonalization of the adjacency matrix. The specific construction allows for easy computation of the resulting blocks. The Gershgorin Circle Theorem is then used to compute the Ramanujan bounds on the spectra.