  We have established a method to measure picograms-per-gram (pg g$^{-1}$)
levels of $^{238}$U and $^{232}$Th in scintillation films by combining the dry
ashing method and inductively coupled plasma mass spectrometry. Trace amounts
of $^{238}$U and $^{232}$Th were measured in up to 2~g of the scintillation
film with almost 100% collection efficiency. This paper details the
experimental procedure, including the pretreatment of the samples and labware,
detection limit of the method, collection efficiencies of $^{238}$U and
$^{232}$Th, and measurement of $^{238}$U and $^{232}$Th in a polyethylene
naphthalate film. This method is also applicable to $^{238}$U and $^{232}$Th
measurements in other low-background organic materials for rare event search
experiments.
