Although mulberries can be propagated by cuttings, it seems much easier (to me) to grow one from seed. According to what I've read, however, it takes 10 years to get fruit from a tree grown from seed.
Washington State U. has a somewhat detailed page about propagation of trees from cuttings. You'll find the mulberry information on the chart down near the end of the page.
I also read that black mulberries are more difficult to propagate from cuttings, than are the white and red types, so the particular tree in question may make a difference as to whether you can successfully clone it.
[url=https://cru.cahe.wsu.edu/CEPublications/pnw0152/pnw0152.html]WSU tree propagation page[/url]
[url=https://www.crfg.org/pubs/ff/mulberry.html]CA Rare Fruit Growers page about mulberry[/url]