We construct a minimal dynamical system of mean dimension equal to $1$, which can be embedded in the shift action on the Hilbert cube $[0,1]^mathbb{Z}$. Our result clarifies a seemingly plausible impression and finally enables us to have a full understanding of (a pair of) the exact ranges of all possible values of mean dimension, within which there will always be a minimal dynamical system that can be (resp. cannot be) embedded in the shift action on the Hilbert cube. The key ingredient of our idea is to produce a dense subset of the alphabet $[0,1]$ more gently.
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