I am planning a trip around the world this fall and want to take a
cellphone that will work in Europe (England, Belgium, Italy), Africa
(mainly South Africa)Dubai, India, Singapore, Bejing, and
Tokyo.......Is there anything I can rent for the six week period??

Hello!

I have personally used WorldCell:
http://www.worldcell.com

Their international phone rental program described here:
http://www.worldcell.com/international.html
lists every country you specify in your question. Their pricing can be
seen by clicking the "Rates" link at the top of the frame. Their
service areas are in the "Coverage" section.

That was the easy answer - This is where it gets more complicated,
however.
 
Many countries around the world, and indeed all but one country you
specified in your post, use the GSM (Global System for Mobile
Communications) standard. This means, that you would be able to use
just one phone (and associated "SIM Chip") in every country you will
be visiting. However, one country, Japan, operates on the "PDC" (or
Personal Digital Cellular) standard, which means that you would not be
able to use a rented GSM phone for your entire Journey.

For a better description of the various cell phone standards around
the world, visit:
http://kropla.com/mobilephones.htm

Essentially what this means is that you would need to rent two phones
from Worldcell (or indeed any other cell phone rental agency) - One
GSM phone, and one for Japan. My recommendation would be to call
Worldcell at 1-888-Worldcell and see if they can be of any assistance.
They do rent GSM and PDC phones.

Other companies providing the same service as WorldCell are:

Global Phone Works
http://www.globalphoneworks.com

Planet Fone
http://www.planetfone.com

Action Cellular:
http://www.globalphone.net

UniverCell:
http://www.rentalcell.com

NexTel also seems to rent their phones, but their page has no mention
of Tokyo or Japan:
http://www.nextel-rental.com/

Search terms I used:
worldcell
international mobile phone rental
international cell phone

Have a great and safe journey!

Axe

Thank you for the answer AND for the people that sent comments.  I am
very pleased with the information.

Nextel Rental seems to have what you need for $10 a day.

Try http://www.nextel-rental.com/ .

Nextel phones apparently don't work in Japan or the United Arab Emirates.

Since Europe and many countries use GSM (Global System for Mobile
Communications), you may consider buying a multiband phone or
borrowing one and use the SIM card from your phone on the loaner (so
the bill comes to your account). On the country where your multiband
phone does not work, you can look to rent. Generally, rental fees are
outrageous. I am going to Europe with such a loaner.

You could also try www.wirelessglobalservices.com or call them at
1-888-872-ROAM (7626) and they will deliver the phone to you within 24
hours to your departure on a guaranteed credit card. Works in up 120
countries. When you check out this site it even details usage charges.
Have a great trip!!