What careers might one be able to ensue upon completing a Bachelors in Arts with a focus in Art History?

You could be a consultant to movie makers for historical films and documentaries. You could be a museum investigator. You could be a consultant to architects and interior designers. You could be a museum curator. You could be a private investigator for private collectors. You could be a tutor to students of private schools’ art programs. You could be an instructor in public schools’ art programs. You could be a product specialist for mail order catalogs offering reproduction art for home enhancement. You could design packaging for Department stores products to suggest their higher tone and value. You could be the designer of web sites about Art History. You could be the consultant to the designer of Art History web sites. You could be the stamp selection specialist for the U.S. Postal Service’s Christmas collection.

You can’t be a professor with just a BA, you would need to continue your education.

Antiquarian Book Trade Antiques Dealer Architectural Conservation Art Advisor Art Consultant: Hotels Art Gallery Art Investment
Artist Representative Art Law and Law Enforcement Art Librarian Arts Organization Consultant Corporate Curator Curatorial Consultant
Estate and Art Appraiser Freelance Collection Manager Freelance Writing Governmental Agencies Independent Producer: Film & TV Museum Work
Preservation and Conservation Publishing Teaching Visual Resource Curator

Source

Gallery director, Teacher, Museum Curator

Aside from trying to get some experience in a gallery, as @JLeslie said, you really need to continue your education if you want a job specifically related to what you’ve studied. Where I’m living, a degree isn’t worth much at all now.
You could always do a hDip and try to get a job teaching