  Let $\mu$ be a finite positive Borel measure on the interval $[0,1)$ and
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ce\`{a}sro-like
operator is defined by $$
\mathcal{C}_\mu(f)(z)=\sum^\infty_{n=0}\mu_n\left(\sum^n_{k=0}a_k\right)z^n, \
z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment of
the measure $\mu$, that is, $\mu_n=\int_{[0, 1)} t^{n}d\mu(t)$. In this paper,
we characterize the measures $\mu$ for which $\mathcal{C}_\mu$ is bounded
(compact) from one Bloch type space, $\mathcal {B}^{\alpha}$, into another one,
$\mathcal {B}^{\beta}$.
