  Given two complex Hilbert spaces $H$ and $K$, let $S(B(H))$ and $S(B(K))$
denote the unit spheres of the C$^*$-algebras $B(H)$ and $B(K)$ of all bounded
linear operators on $H$ and $K$, respectively. We prove that every surjective
isometry $f: S(B(K)) \to S(B(H))$ admits an extension to a surjective complex
linear or conjugate linear isometry $T: B(K)\to B(H)$. This provides a positive
answer to Tingley's problem in the setting of $B(H)$ spaces.
