Let $H^\infty$ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by $\mathscr{ B}(H^\infty)$ the Banach space of all bounded linear operators from $H^\infty$ to itself. We prove that the Bishop-Phelps-Bollob\'{ a}s property holds for $\mathscr{ B}(H^\infty)$. As an application to our approach, we prove that the Bishop-Phelps-Bollob\'{ a}s property also holds for operator ideals of $\mathscr{ B}(H^\infty)$.
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