We describe a hybrid vacuum system in which a single ion or a well-defined small number of trapped ions (in our case Ba or Rb) can be immersed into a cloud of ultracold neutral atoms (in our case Rb). This apparatus allows for the study of collisions and interactions between atoms and ions in the ultracold regime. Our setup is a combination of a Bose-Einstein condensation apparatus and a linear Paul trap. The main design feature of the apparatus is to first separate the production locations for the ion and the ultracold atoms and then to bring the two species together. This scheme has advantages in terms of stability and available access to the region where the atom-ion collision experiments are carried out. The ion and the atoms are brought together using a moving one-dimensional optical lattice transport which vertically lifts the atomic sample over a distance of 30 cm from its production chamber into the center of the Paul trap in another chamber. We present techniques to detect and control the relative position between the ion and the atom cloud.