Frequent handovers between satellites, and the lack of a central (on-board) state database, challenge the delivery of communication services from Low-Earth-Orbit (LEO) satellite constellations. Traditional blockchain protocols to maintain distributed state are unsuitable for on-board deployment as they do not take the moving dynamics and inter-satellite links of constellation orbits into account. In this paper, we propose LeoDist, a distributed ledger for LEO constellations. We introduce novel concepts, such as leader-handover, neighbor synchronization, and service area gossiping to meet the challenges of this dynamic environment. To the best of our knowledge, LeoDist is the first distributed ledger on board LEO satellites to support the core routing, state synchronization and consensus blockchain protocols while accounting for orbital dynamics and coverage area. We show using a lab testbed with the NASA core Flight System (cFS) that LeoDist is able to route, gossip, reach consensus, synchronize state and process transactions efficiently, offering 3-4 orders of magnitude faster leader failover and handover compared to Raft, up to 171 distributed transactions per second, and broadcasting across a constellation of 95 satellites in less than 50ms on standard UDP/IP links.