Let B be any intermediate closed algebra between \(H^{\infty}\) and \(L^{\infty}({\mathbb{T}})\), i.e. B is a Douglas algebra. The algebra \(Q_ B=\bar B\cap B\) is closely related to B and admits a description in terms of vanishing mean oscillation. One can consider the algebra \(QA_ B=H^{\infty}\cap \bar B\) which consists of the functions in \(Q_ B\) which can be extended analytically to \({\mathbb{D}}\). If \(B=L^{\infty}\) then clearly \(QA_ B=H^{\infty}.\) Therefore it looks quite natural to investigate \(QA_ B\) along the same line as it has been made in case of \(H^{\infty}.\) The paper deals with the interpolation problem for \(QA_ B\). Let \({\mathfrak M}(B)\) be the maximal ideal space of B. We say that \(\{z_ n\}\subseteq {\mathbb{D}}\) is thin near \({\mathfrak M}(B)\) if \(\inf_{n}\prod_{m\neq n}| \frac{z_ m-z_ n}{1-\bar z_ mz_ n}|>0,\) i.e. \(\{z_ n\}\) is an interpolating sequence for \(H^{\infty}\), and \(\prod_{m\neq n}| \frac{z_ n-z_ m}{1-\bar z_ mz_ n}| \to 1\) as \(z_ n\) approximates \({\mathfrak M}(B)\). Theorem. The following conditions are equivalent for a sequence \(\{z_ n\}\subseteq {\mathbb{D}}:\) (1) for any bounded sequence of complex numbers \(\{\lambda_ n\}\) there exists \(f\in QA_ B\) such that \(f(z_ n)=\lambda_ n\) for all n; (2) for any bounded sequence of complex numbers \(\{\lambda_ n\}\) there exists \(f\in VMO_ B\) such that \(f(z_ n)=\lambda_ n\) for all n; (3) \(\{z_ n\}\) is thin near \({\mathfrak M}(B)\). Moreover if (3) holds, then there exists P. Beurling functions yielding (1). That is, there are functions \(\phi_ n\in QA_ B\) such that \(\phi_ n(z_ k)=\delta_{nk}\) and for any bounded sequences \(\{\lambda_ n\}\), \(\sum_{n}\lambda_ n\phi_ n\in QA_ B.\)