Don't know about morus nigra, but believe most of the white mulberries (morus alba) sold for landscape use in DFW are fruitless cultivars, so invasiveness is not an issue with these. Very popular in the late 1960s when our neighborhood was developed. We enjoyed the two large ones we inherited. Good if you want a large/broad tree relatively quickly, but otherwise there are many better alternatives for north Texas.

Here is a link that might be useful: Morus alba Fruitless Cultivars: White Mulberry

True Morus nigra resembles a common fig tree and grows about the same size. It is seen in places like Britain and the Pacific Northwest - perhaps it may not even survive in your area, let alone become weedy.