Originally Posted by Texas Star

As can be seen here, the height of the mammoth would make taking the frontal brain shot pretty hard. That would be a real problem in a charge.

In colder weather, these animals had a long, shaggy coat that might also impede bullet penetration.

But our ancestors killed them with spears and darts from atlatls. The famous Clovis points from New Mexico were probably on atlatl darts. In some areas, mainly Eastern Europe, houses were made of mammoth tusks and bones. These were probably overlaid with mammoth hides. Some may have had mud overlaying the hides and bones.
My guess is that anything from a .375 H&H Magnum to a .700 Nitro Express would kill a mammoth, and the .340 Weatherby might suffice, too. You could probably kill them with a .30/06, but I'd hate to stop one with anything below .375 caliber. These were bigger than a modern elephant. It's illegal in most African countries to hunt elephant with less than .375 caliber, in some cases, .400 caliber. But keep in mind your own recoil limitations.

Better start looking for a nice .416 Rigby now, or have it built.