The 19-year-old striker has agreed a loan deal which will see him remain at St Andrews until January 12th, with an option for that to be extended.
The England Under 19 international has previously enjoyed loan stints at Milton Keynes Dons and Oxford United.
Birmingham boss Lee Clark was delighted, saying: "Rob is a very quick-footed attacker who has got great ability to go past defenders and creates great opportunities for himself and others.
"He's also got a great strike of the ball so I'm looking forward to working with him and he's excited about the challenge.
"It was quite a coup getting Rob because he's been doing really well in the development squad at West Ham which has got him on the bench for most of the season in the Premier League.
"I have got to thank Sam Allardyce for allowing this one to happen. In all honesty, I didn't think he'd allow Rob to come as he's been a big part of their squad but thankfully he has. We're now looking forward to working with him."