Summary: For any \(m\)-input, \(m\)-output, finite-dimensional, linear, minimum-phase plant \(P\) with first Markov parameter having spectrum in the open right half complex plane, it is well known that the adaptive output feedback control \(C\), given by \(u=-ky\), \(\dot k= \|y\|^2\), yields a closed-loop system \([P,C]\) for which the state converges to zero, the signal \(k\) converges to a finite limit, and all other signals are of class \(L^2\). It is first shown that these properties continue to hold in the presence of \(L^2\)-input and \(L^2\)-output disturbances. Working within the conceptual framework of the nonlinear gap metric approach to robust stability, and by establishing gain function stability of an appropriate closed-loop operator, it is proved that these properties also persist when the plant \(P\) is replaced with a stabilizable and detectable linear plant \(P_1\) within a sufficiently small neighborhood of \(P\) in the graph topology, provided that the plant initial data and the \(L^2\) magnitude of the disturbances are sufficiently small. Example 9 of Georgiou and Smith [IEEE Trans. Automat. Control, 42, 1200–1221 (1997; Zbl 0889.93043)] is revisited. Unstable behavior for large initial conditions and/or large \(L^2\) disturbances is shown, demonstrating that the bounds obtained from the \(L^2\) theory are qualitatively tight: this contrasts with the \(L^\infty\)-robustness analysis of Georgiou and Smith, which is insufficiently tight, to predict the stable behavior for small initial conditions and zero disturbances.