If you browse the US Postal Services website, you will see that "frank" applies to mail that is sent free-of charge, i.e. by members of Congress who have "franking priviledges".

http://pe.usps.gov/search/jsp/search/vv_docread.jsp?k2dockey...

From the site:

6.0 Official Mail (Franked)
6.1 Basic Information
6.1.1 Members of Congress
Official mail of Members of Congress is sent without prepayment of postage and bears instead either a written or printed facsimile signature or a specified marking. Exhibit 6.1.1 shows what is accepted under frank and who is authorized its use.

Exhibit 6.1.1 Franked Mail of Members of Congress

"Stamping" also sounds wrong me, I would use something like "determine postage costs" - but you can certainly find a better term from the above website.

¨Franking¨ is almost exclusively used in reference to privileges held by US members of congress. I´d say stamping is much more common.

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Note added at 19 mins (2012-05-24 07:27:43 GMT)
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Postage would be even more common, but only in reference to the amount charged. The actual physical act of applying a stamp (as I read the example) would commonly be ¨stamping¨.

I would say "placing stamps on envelopes," (unless it is a Congressional office or other governmental office that does not use stamps).

Unless of course from the context of the text (not provided), which might imply the use of a postage meter (rather than stick on stamps). In which case, "affixing postage with the postage meter" would be more accurate.

This is not what I was expecting, but it seems to be what it's called.

First, I assume we are talking here about the system business and organisations use to frank mail using a franking machine (as we call it in the UK), printing a mark on the envelope to show the postage paid, as opposed to sticking a stamp on the envelope.

As far as I can see, the situation is that this can be called "franking" in the US, but in practice it isn't. Merriam Webster defines the verb "frank" as:

"1 a : to mark (a piece of mail) with an official signature or sign indicating the right of the sender to free mailing
b : to mail free
c : to affix to (mail) a stamp or a marking indicating the payment of postage"
http://www.merriam-webster.com/dictionary/frank

American Heritage defines it the same way. Meaning 1 (a) presumably applies to members of Congress, as mentioned here, and seems to be common usage. Meaning 1 (c) is the one we want, and there it is. But it doesn't seem to be current. Note also that "franking", according to this, covers not only doing it with franking machine but also sticking a stamp on. This is confirmed here:
http://en.wikipedia.org/wiki/Franking

In the UK, however, we don't use the term "franking" to apply to postage stamps, but only the business machine thing, or alternatively cancelling the postage stamp with a postmark. Collins (which is UK-orientated) defines it like this:

"mainly British to put a mark on (a letter, parcel, etc), either cancelling the postage stamp or in place of a stamp, ensuring free carriage"
http://www.collinsdictionary.com/dictionary/english/frank

What about "stamping"? The American dictionaries referred to define it exclusively (in the context of mail) as sticking a stamp on, with no reference to what we call franking in the UK. Here's Merriam Webster again:

"3 b : to attach a stamp to"
http://www.merriam-webster.com/dictionary/stamp

So I don't think "stamping" is the US equivalent of "franking". It's true that Arthur Pitney called his franking machine (as we would call it in the UK) a "stamping machine", as Tony has pointed out, but that was in 1902, and nowadays the term is not in general use.

It's called metering. The US Postal Service says that "A clear space must be available on all mail for the address, postage (permit imprint, postage stamp, or meter stamp), postmarks, and postal endorsements."
http://pe.usps.com/text/dmm300/102.htm

Here's an explanation:
"Difference Between Metered & Stamped Mail
Metered mail is widely used by businesses and non-profits who have bulk mailings. The cancelation and postage information is printed directly onto the piece of mail or a sticker by a postage meter.
Postage meters are available for purchase by five authorized manufacturers in the US: Pitney Bowes, Data-Pac Mailing Systems, FP Mailing Solutions, Hasler Inc. and Neopost. In addition to purchasing the meter and accessories, a business or non-profit must apply for a metered mail permit."
http://www.ehow.com/about_5478782_difference-between-metered...

If you look, you'll find plenty of references to postage meters, metering mail, metered mail and so on. But "stamped", as in the source just cited, means it has an adhesive postage stamp attached, and American references to "frank"/"franking", in this sense, are pretty few and far between.

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Note added at 1 hr (2012-05-24 08:51:46 GMT)
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It's either "franking" or "metering", that's for sure, and Jenni's experience carries weight. But others here seem to find "franking" unfamiliar in this sense, and both the USPS and many other places refer to "metering". Google results for "franking" are overwhelming UK. So if it's specifically US usage you want, "metering" looks like the best bet to me. Ideally we need to hear from more Americans.

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Note added at 3 hrs (2012-05-24 11:05:24 GMT)
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When I said "printing a mark on the envelope" near the beginning, what usually happens is the the franking machine or postage meter prints a label which you stick on the envelope where the stamp would go, but the terminology's the same.