We investigate the problem of multi-party private set intersection (MP-PSI). In MP-PSI, there are $M$ parties, each storing a data set $\mathcal{p}_i$ over $N_i$ replicated and non-colluding databases, and we want to calculate the intersection of the data sets $\cap_{i=1}^M \mathcal{p}_i$ without leaking any information beyond the set intersection to any of the parties. We consider a specific communication protocol where one of the parties, called the leader party, initiates the MP-PSI protocol by sending queries to the remaining parties which are called client parties. The client parties are not allowed to communicate with each other. We propose an information-theoretic scheme that privately calculates the intersection $\cap_{i=1}^M \mathcal{p}_i$ with a download cost of $D = \min_{t \in \{1, \cdots, M\}} \sum_{i \in \{1, \cdots M\}\setminus {t}} \left\lceil \frac{|\mathcal{p}_t|N_i}{N_i-1}\right\rceil$. Similar to the 2-party PSI problem, our scheme builds on the connection between the PSI problem and the multi-message symmetric private information retrieval (MM-SPIR) problem. Our scheme is a non-trivial generalization of the 2-party PSI scheme as it needs an intricate design of the shared common randomness. Interestingly, in terms of the download cost, our scheme does not incur any penalty due to the more stringent privacy constraints in the MP-PSI problem compared to the 2-party PSI problem.