Not a fan of M.alba, and have mostly refrained from purposely planting or grafting cultivars that I know are pure M.alba, though I have grafted some M.rubra and hybrid selections onto M.alba understock. But... most of the really good fruiting cultivars that work in most of the USA are hybrids of M.rubra X alba. Think "Illinois Everbearing', 'Silk Hope', 'Collier', 'Stearns', 'Lawson Dawson', etc.
And... trashy M.alba is present in every abandoned fencerow or ditchbank in town, 5 miles away... don't think for a minute that birds won't feast in town and poop out seeds here on the farm... so, as I said earlier... that cat is already out of the bag. Even though our native M.rubra is still pretty common out here in the country, KY Div. of Forestry has abdicated on trying to find pure M.rubra... I bought a bundle of 100 'Red mulberry' from the state forestry nursery, a few years back... not a pure M.rubra in the bunch... all are hybrids, and most look like they're probably 3/4 alba 1/4 rubra, at best. Had a discussion on that with the local KY forestry guy just recently... he said that, in the past, he'd been the one assigned to gather mulberry seed. He'd given up on finding fruiting females (I know where at least two big ones are, right next to the road)... so just gathered fruit from any fruiting mulberry they could find.
There's a lot of confusion/misinformation about mulberries in the 'nursery trade'... a lot of varieties are labeled 'Black mulberry' or M.nigra (the TRUE Black mulberry)... but are not... they're just purple/black-fruited selections... in many cases, a M.alba, M.macroura, or an albaXrubra hybrid True M.nigra/Black mulberry will not survive in the vast majority of the US... it prefers a 'Mediterranean' climate... will not tolerate the humidity of the South/Southeast and Midwest.
Lucille Whitman offers a couple of true M.nigras - 'Noir de Spain' and 'King James'... but one that she was advertising as M.nigra, a few years ago, 'Sicilian Black' is not... it's a M.alba.
Unless your 'Black Beauty' is coming from a west-coast nursery, like Dave Wilson Nurseries or Whitman Farms... it won't be a M.nigra, regardless of what the nursery says...