Let \(\Omega\) be an open subset of \(\mathbb{R}^n\) and \(\text{MBO}(\Omega)\) denote the space of functions in \(L^1_{\text{loc}}(\overline\Omega)\) with bounded mean oscillation. The authors’ main result is that if \(\Omega\) is sufficiently regular, then there is a bounded linear extension operator from \(\text{BMO}_1(\Omega)\) into \(\text{BMO}_1(\mathbb{R}^n)\), where \(\text{BMO}_1(\Omega)\) is somewhat smaller than \(\text{BMO}(\Omega)\). An earlier extension theorem was obtained by P. W. Jones [Indiana Univ. Math. J. 29, 41-66 (1980; Zbl 0432.42017)] with a different regularity condition on \(\Omega\) and a different subset of \(\text{BMO}(\Omega)\).