I am seeking an online payment service where I can use a Visa or
MasterCard to make a payment to another individual or business located
in the United States. I am looking for a service that will not charge
a fee to either myself or the recipient of the funds. The service
needs to allow me to make payments which total at least $1,200 in any
given month. In addition, I need to be able to make payments of at
least $300 per transaction. In other words, if the service limits my
payment per transaction to $20, this is not a suitable solution.

Ideally the charges that appear on my credit card as a result of the
service will not be categorized as "cash advances." However I am aware
that this additional criteria may be difficult to determine without
knowing the terms of my credit card provider.

Simple; Have you tried http://www.paypal.com You get $5 for signing
up, you also get money for any referrals.

PayPal charges a fee to receive credit card payments. As I mentioned
in the question, I am looking for service that does not charge a fee.

marc10,

As MasterCard and VISA both charge a fee (a small flat amount plus a
percentage of the transaction amount) for every transaction, I do not
think that you will be able to find such a service.

Good Luck!

aceresearcher

c2it a service run by Citibank, appears to meet the your requires. 
Although your credit card compnay may charge you cash advance fees. 
However if you use it with a normal bank acount instead of a credit
card then there are no fees.

https://www.c2it.com

A quote from there faq follows

With c2it you can send, receive and move money within the U.S. for
free.*
* We will not charge a transaction fee when you send cash to someone
within the U.S. using your bank account or credit card. If you have
established a c2it line of credit, you will incur finance charges if
you choose to fund your transaction from that credit line. Transaction
fees will apply for sending money to someone outside the U.S. Your
credit card company may assess a finance charge and fee each time you
send money.

popsracer wrote that Citibank's c2it service is free. Yes, it is right
now, but only for 8 more days. As the site discloses to current
members (but not to anyone else), "We're changing our service. In
order to continue providing you with our high level of service, c2it
is making changes to the domestic Send Cash fee and Add Cash
functionality effective February 23, 2003. By using the service after
this date, you're agreeing to these new Terms and Conditions."
Briefly, in 8 days it will cost 2% of the transaction amount to send
money, with a minimum fee of 30 cents per transaction.

The timing of C2it initiating fees and the posting of this question is
not a coincidence. I have been using the service to fulfill my needs
for about the past year and now looking for another solution. Previous
to C2it.com, I was using a service called Payme.com and prior to that
I was using BillSender.com. All of these services matched the criteria
I outlined above...and all of these services are either no longer in
business or have begun to charge fees. I am aware of the economics of
these services and never understood how they made money, but I was
happy to use them. I still think there may be one out there...