So with help from Ian and Anthony it looks like my comment above can be turned into an answer. To summarize:

Let S be the substitution on two symbols given by $0\to001, 1\to11100$. Let $x\in\{0,1\}^{\mathbb N}$ be the limit of $S^n(01)$. Let $\Sigma$ be the orbit closure of $x$ under the shift map. 

$(\Sigma,\sigma)$ is uniquely ergodic and weak mixing but not strong mixing (Dekking and Keane '83). Let $m$ be the invariant measure. We have that $m[0]=m[1]=\frac{1}{2}$, just by looking at the leading eigenvalue of the matrix (22|13).

Furthermore, primitive substitution systems have zero entropy. I don't know the original reference for this, but much stronger results are known, for example that the complexity function of any primitive substitution has sublinear growth. Apparently this is proved in J.-J. PANSIOT: Complexite des facteurs des mots innis enegendres par morphismes iteres, 1984, or in Queffelec 'substitution dynamical systems'. I read it as a comment on page 5 of Ferenczi 'complexity in sequences and dynamical systems'.

Thus the system $(\{0,1\}^{\mathbb N},\sigma,m)$ satisfies the requirements of your proposition.