We prove that a one-dimensional Noetherian domain is slender if and only if it is not a local complete ring. The latter condition for a general Noetherian domain characterizes the domains that are not algebraically compact. For a general Noetherian domain R we prove that R is algebraically compact if and only if R satisfies a condition slightly stronger than not being slender. In addition we enlarge considerably the number of classes of rings for which the question of slenderness can be answered. For instance we prove that any domain, not a field, essentially of finite type over a field is slender.