  A Kerr-de Sitter black hole is a solution
$(M,g_{\Lambda,\mathfrak{m},\mathfrak{a}})$ of the Einstein vacuum equations
with cosmological constant $\Lambda>0$. It describes a black hole with mass
$\mathfrak{m}>0$ and specific angular momentum $\mathfrak{a}\in\mathbb{R}$. We
show that for any $\epsilon>0$ there exists $\delta>0$ so that mode stability
holds for the linear scalar wave equation
$\Box_{g_{\Lambda,\mathfrak{m},\mathfrak{a}}}\phi=0$ when
$|\mathfrak{a}/\mathfrak{m}|\in[0,1-\epsilon]$ and
$\Lambda\mathfrak{m}^2<\delta$. In fact, we show that all quasinormal modes
$\sigma$ in any fixed half space $\Im\sigma>-C\sqrt\Lambda$ are equal to $0$ or
$-i\sqrt{\Lambda/3}(n+o(1))$, $n\in\mathbb{N}$, as
$\Lambda\mathfrak{m}^2\searrow 0$. We give an analogous description of
quasinormal modes for the Klein-Gordon equation.
  We regard a Kerr-de Sitter black hole with small $\Lambda\mathfrak{m}^2$ as a
singular perturbation either of a Kerr black hole with the same angular
momentum-to-mass ratio, or of de Sitter spacetime without any black hole
present. We use the mode stability of subextremal Kerr black holes, proved by
Whiting and Shlapentokh-Rothman, as a black box; the quasinormal modes
described by our main result are perturbations of those of de Sitter space. Our
proof is based on careful uniform a priori estimates, in a variety of
asymptotic regimes, for the spectral family and its de Sitter and Kerr model
problems in the singular limit $\Lambda\mathfrak{m}^2\searrow 0$.
