I can't beleive that no one teaches how to multiply by 1 anymore. If you can do that, you can convert any units without difficulty.
For your problem you need to know this:
[*]7.8*10^8 km^3 = 7.8*10^8 km^3 * 1 * 1 * 1 * 1
If you can do that, you can answer your problem without google, and without help from others. In fact, you can use this for all of your other problems.
How? Well that is easy. There are 100,000 cm in 1 km. Or in other words:
1 = (100,000 cm / 1 km).
Now use this in the bulleted equation above:
[*]7.8*10^8 km^3 = 7.8*10^8 km^3 * (100,000 cm / 1 km) * (100,000 cm / 1 km) * (100,000 cm / 1 km)
Notice how the kilometers all cancel out? You are left with:
[*]7.8*10^8 km^3 = 7.8*10^23 cm^3.
Now use minendo's version of 1:
1 = (1 mL / 1 cm^3).
[*]7.8*10^8 km^3 = 7.8*10^23 cm^3 * 1 = 7.8*10^23 cm^3 * (1 mL / 1 cm^3) = 7.8*10^23 mL.