I think the mole is defined from the number of
attons in a certain mass
 of a certain isotope of an certain element (carbon 12 IIRC)
 
 The mole is the amount of a substance which contains Avogadro's number=20
 (6.0225 =D7 10^23 ) of atoms or molecules of the substance. 
 
Sure. Defining the mole is equivalent to defining Avogadro's number. But
that's similar to saying that as the electonic charge is 1.602*10^-19C, a
colomob is the charge on 6.24*10^18 electrons. so defining the coulomb
is equivalent to definint that number
  Stated differently, it is the same amount of grammes
of the substance as=20
 that element's molecular weight.
 The definition says that it is the amount of a substance which contains=20
 the same number of molecules of the substance as 12 grammes of=20
 carbon-12, which is a way of expressing the number of molecules in such=20 
I have an idea that the definition I came across specifies Carbon-12.
Since the atomic masses are not exact integer rations of each other (in
other words, an oxygen-16 atom is not exactly 4/3 of the mass of a
carbon-12 atom), I think you do have to specify the element you are using
here.
-tony