We study the metastable dynamics of a discretised version of the mass-conserving stochastic Allen-Cahn equation. Consider a periodic one-dimensional lattice with $N$ sites, and attach to each site a real-valued variable, which can be interpreted as a spin, as the concentration of one type of metal in an alloy, or as a particle density. Each of these variables is subjected to a local force deriving from a symmetric double-well potential, to a weak ferromagnetic coupling with its nearest neighbours, and to independent white noise. In addition, the dynamics is constrained to have constant total magnetisation or mass. Using tools from the theory of metastable diffusion processes, we show that the long-term dynamics of this system is similar to a Kawasaki-type exchange dynamics, and determine explicit expressions for its transition probabilities. This allows us to describe the system in terms of the dynamics of its interfaces, and to compute an Eyring-Kramers formula for its spectral gap. In particular, we obtain that the spectral gap scales like the inverse system size squared.