  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 >3sigma 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 E-3 per cubic parsec 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.
