  We prove that for $1<c<4/3$ the subsequence of the Thue--Morse sequence
$\mathbf t$ indexed by $\lfloor n^c\rfloor$ defines a normal sequence, that is,
each finite sequence $(\varepsilon_0,\ldots,\varepsilon_{T-1})\in \{0,1\}^T$
occurs as a contiguous subsequence of the sequence $n\mapsto \mathbf
t\left(\lfloor n^c\rfloor\right)$ with asymptotic frequency $2^{-T}$.
