Directed acyclic graphs (DAGs) are emerging as an attractive alternative to traditional blockchain architectures for distributed ledger technology. In particular, DAG ledgers with stochastic attachment mechanisms potentially offer many advantages over blockchain, including scalability and faster transaction speeds. However, the random nature of the attachment mechanism coupled with the requirement of protection against double-spending transactions might result in an unstable system in which not all transactions get eventually validated. Such transactions are said to be orphaned, and will never be validated. Our principal contribution is to propose a simple modification to the attachment mechanism for the Tangle (the IOTA DAG architecture). This modification ensures that all transactions are validated in finite time, and preserves essential features of the popular Monte Carlo selection algorithm. In order to demonstrate these results, we derive a fluid approximation for the Tangle (in the limit of infinite arrival rate) and prove that this fluid model exhibits the desired behavior. We also present simulations that validate the results for finite arrival rates.