One of the great things about going on safari is the opportunity to buy a new rifle, get proficient with it and enjoy a completely new experience. Why not consider a bolt action 375 (pick your flavor) that shoots one kind of bullet into tiny-tiny holes? Swift A-Frame, Norma Oryx, Barnes TTSX/TSX, Federal TBBC...pick a premium expanding bullet from what the above members have suggested and you will be in excellent shape.
Seriously, I'm disappointed in the membership here that it's taken until post #20 to come up with this possibility.
@Rubberhead - Welcome to a world of enablers.
BTW - Do not underestimate the toughness of both the Zebra and Blue Wildebeest. Do your homework and check out the "shot placement" section. No points get deducted for using a larger caliber.