Multi-Agent Meeting (MAM) is the problem of finding a meeting location for multiple agents and paths to that location. Recently, a Multi-Directional Heuristic Search algorithm, called MM*, was introduced. MM* is a state-of-the-art MAM optimal solver that searches from multiple directions (one for each agent) and is guided by a heuristic function. Practically, a solution to MAM may contain conflicting paths. A related problem that plans conflict-free paths to a given set of goal locations is the Multi-Agent Path Finding problem (MAPF). In this paper, we solve the Conflict-Free Multi-Agent Meeting problem (CF-MAM). In CF-MAM, we find a meeting location for multiple agents (as in MAM) as well as conflict-free paths (as in MAPF) to that location. We introduce two novel algorithms, which combine MAM and MAPF solvers, for optimally solving CF-MAM. We compare both algorithms experimentally, showing the pros and cons of each algorithm.