Traditionally, infinite databases were studied as a data model for queries that may contain function symbols (since functions may be expressed as infinite relations). Recently, the interest in infinite databases has been sparked by additional scenarios, e.g., as a formal model of a database of an openworld software or of other relations that may be spread across the Web. This paper revisits the weak safety and termination problems for recursive DATALOG programs evaluated over infinite databases. In particular, an algorithm is presented that computes all finiteness constraints for the IDB predicates of a program, given a set of finiteness constraints over the EDB predicates. In addition to being of interest in itself, this algorithm also presents an alternative method to check for weak safety and as a skeleton for query evaluation. A sufficient condition for program termination is also presented, provided that the program and database satisfy certain natural constraints. These constrai...