Adding a finite set to a non-regular language

Suppose $$A = \{0^{n}1^{n} \mid n \ge0\}$$, which is not regular, and let $$B$$ be a finite subset of $$\Sigma^* \setminus A$$. Is $$A \cup B$$ regular?

Note the closure properties of regular languages. What is $$(A \cup B) \setminus B$$? What does that imply about $$(A \cup B)$$ since we know $$B$$ is regular (due to being finite)?