We identify proper motion objects in the Hubble Ultra Deep Field (UDF) using the optical data from the original UDF program in 2004 and the near-infrared data from the 128 orbit UDF 2012 campaign. There are 12 sources brighter than I = 27 mag that display >3σ significant proper motions. We do not find any proper motion objects fainter than this magnitude limit. Combining optical and near-infrared photometry, we model the spectral energy distribution of each point-source using stellar templates and state-of-the-art white dwarf models. For I ≤ 27 mag, we identify 23 stars with K0-M6 spectral types and two faint blue objects that are clearly old, thick disk white dwarfs. We measure a thick disk white dwarf space density of 0.1-1.7 × 10–3 pc–3 from these two objects. There are no halo white dwarfs in the UDF down to I = 27 mag. Combining the Hubble Deep Field North, South, and the UDF data, we do not see any evidence for dark matter in the form of faint halo white dwarfs, and the observed population of white dwarfs can be explained with the standard Galactic models.