We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses $x$ and $y$ coalesces at a given rate $K(x,y)$. A particle of mass $x$ fragments into a collection of particles of masses $θ\_1 x, θ\_2 x, \ldots$ at rate $F(x) β(dθ)$. We assume that the kernels $K$ and $F$ satisfy Hölder regularity conditions with indices $λ\in (0,1]$ and $α\in [0, \infty)$ respectively. We show existence of such infinite particle systems as strong Markov processes taking values in $\ell\_λ$, the set of ordered sequences $(m\_i)\_{i \ge 1}$ such that $\sum\_{i \ge 1} m\_i^λ \textless{} \infty$. We show that these processes possess the Feller property. This work relies on the use of a Wasserstein-type distance, which has proved to be particularly well-adapted to coalescence phenomena.