You will be required to pass thro Immigration .The Us Customs will come on board you will need to complete the Green Visa waiver Form etc.I don't know if you are aware you will also be required to apply for ESTA clearance this must be completed before you travel . This can be done on line.If you have any doubt perhaps Cunards visa section could help.
Sorry to have to tell you this but not only will you have to go through US immigration once more but you should also fill in the ESTA - Electronic System for Travel Authorisation form on the internet and get your own personal number prior to sailing. Just enter ESTA in your usual search engine and follow the instructions it's very simple. The one blessing is that they have allowed US Immigration Officials to keep their humour genes so they are so much more pleasant now. ..... Neil.
Both of the answers already given are correct, but what they didn't tell you was that the US Immigration come on board in the early hours of the morning and you are told to present yourself at an unearthly hour to be checked.
Compared with this the UK Border Service actually get on board in NY and travel with you. You are asked to present yourselves at some point on the way across to Southampton in a very civilised manner.
I know which country I support, even the Americans who travel on the QM2 agree with our method of checking passengers as the info can be electronically transmitted en-route.
I agree with 'Smith' - we crossed from San Juan to Southampton last April and found the arrangement whereby British immigration officers processed all passengers during the final sea day very civilised indeed. We wandered along to one of the smaller bars where they were set up, presented our passports - were asked if we'd enjoyed our cruise and how we'd liked the ship, etc - it all took about five minutes and at the end everyone simply disembarked! When we were told this was going to happen by the cruise director, most of the American passengers gasped in disbelief.