If we assume that(i guess it's about right) a pack of 100 100-dollar bills stacked the one above the other(obviously in the area that is defined by its width and length) has 1 cm height then it's easy to find the volume.
Let's call a bundle, a pack of 100 100-dollar bills. One such bundle has 10000 dollars value.
For 10^9 dollars you need 10^9/10^4 = 10^5 such bundles to use.
Since the volume of one bundle is 15.6·6.6/10^6 m^3, the 10^5 bundles will take (10^5)·15.6·6.6/10^6 m^3 =~ 10.3 m^3
So we will need about 10.3 cubic meters of 100 100-dollar bills to use.
If we want to practically stack them in a room we could stack them in q rectangular cuboid shape of 1 meter height and with 1.56 m x 0.66 m size of its base's sides length.
That means we would use to create the base 10·10=100 bundles and also 100 bundles(1 m height /1 cm height of each bundle) to fill every column of these.
So we would have 10000 such bundles in a rectangular cuboid of size 1m x 0.66m x 1.56m
But we have 100000 bundles so we would need 10 of these rectangular cuboids to create a 1 billion dollar stack.
So 1 billion dollars can be stacked into 10 rectangular cuboids of a size 1m x 0.66m x 1.56m