  In this paper we consider the Riemann--Liouville fractional integral
$\mathcal{N}^{\alpha,\nu}(t)= \frac{1}{\Gamma(\alpha)} \int_0^t
(t-s)^{\alpha-1}N^\nu(s) \, \mathrm ds $, where $N^\nu(t)$, $t \ge 0$, is a
fractional Poisson process of order $\nu \in (0,1]$, and $\alpha > 0$. We give
the explicit bivariate distribution $\Pr \{N^\nu(s)=k, N^\nu(t)=r \}$, for $t
\ge s$, $r \ge k$, the mean $\mathbb{E}\, \mathcal{N}^{\alpha,\nu}(t)$ and the
variance $\mathbb{V}\text{ar}\, \mathcal{N}^{\alpha,\nu}(t)$. We study the
process $\mathcal{N}^{\alpha,1}(t)$ for which we are able to produce explicit
results for the conditional and absolute variances and means. Much more
involved results on $\mathcal{N}^{1,1}(t)$ are presented in the last section
where also distributional properties of the integrated Poisson process
(including the representation as random sums) is derived. The integral of
powers of the Poisson process is examined and its connections with generalised
harmonic numbers is discussed.
