If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as *almost convergent*) iff
$$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L  \quad (*) $$
uniformly in $n$. Setting $n=0$ yields Cesaro summability.

As Aaron says, the converse is false. It was shown by Connor (in his appropriately named article) *Almost none of the sequences of 0's and 1's are almost convergent* that randomly generated sequence of 0's and 1's almost never has property $(*)$ ; but almost every randomly generated sequence of 0's and 1's is Cesaro summable to 1/2 by the law of large numbers.