We study the stability and dynamics of melting icebergs. Specifically, we address the 'toppling' or 'rollover' observed for floating icebergs. The rollover is thought to occur because the ocean melts the iceberg from below, causing its overall mass and mass distribution to change with time. We model the evolution of equilibrium positions for a general homogeneous body afloat in an ideal fluid, as this homogeneous body is subjected to 'melting', i.e. a slow removal of material from the submerged part. If this process is the dominating melting mechanism, can the likelihood of a toppling be inferred from observing only the above-surface part? We show here that some information about the evolution of stability due to melting call be inferred from the surface geometry of the iceberg.