  We show that assuming the generalized Riemann hypothesis there are no normal
CM-fields with class number one of degree 64 and 96. This is done by
constructing complete tables of normal CM-fields using discriminant bounds of
Lee--Kwon. This solves the class number one problem for normal CM-fields
assuming GRH. Using the same technique to solve the relative class number one
problem in degrees 16, 32, 56 and 82, also the corresponding relative class
number one problem is solved assuming GRH.
