mrs_sweetpeach | Can anyone identify this thing?

Last summer I noticed a tree growing along the fence separating my yard from my neighbor's. At the time, it was about 3' tall. It has now reached a height of approximately ten feet, and, last night I noticed that it is bearing fruit. Is this some sort of evil weed tree that will take over my garden if I don't keep an eye out for volunteers?

Am I correct in thinking it is a White Mulberry Tree (Morus alba)?