Do Not Bend Envelope

[SonnyCrockett]

How much does it cost to send it on 1st class within the UK?

Would it need to be weighed at the post office ?

[Buenchico]

I assume that you're referring to the special cardboard-backed envelopes which you can buy with 'Do Not Bend' on them. They have no special status within the Royal Mail system, so their postage is based upon purely upon size and weight as with all other mail.
http://www.royalmail.com/prices-2013#UK%20Standard

[spudqueen]

This is from the Post Office website http://www.postoffice.co.uk/first-class-letters

If it weighs less than 100g (use your kitchen scales) and you're sure it's a small letter then stick an ordinary 1st or 2nd class stamp on. If if weighs less than 100g and you're not sure if it's a small or large letter, or if weighs more than 100g either stick two 1st class stamps or one 1st class and one 2nd class on. 'Twill be ok.