The recipient of the gift won't have to pay Stamp Duty if there's no mortgage on the property but if he/she has agreed to take over an existing mortgage, he/she will then need to pay Stamp Duty if the value of the mortgage is over the threshold:
https://www.gov.uk/guidance/sdlt-transferring-ownership-of-land-or-property
If you're sure that you can understand all of that, then do it yourself. If it's as clear as mud though, pay a solicitor (or licensed conveyancer) to do it!
I can't advise on the cost of using the services of such a professional (sorry!) but I strongly recommend
(a) getting several quotes ; and
(b) if you can find an independent licensed conveyancer (as many seem to have been taken over by solicitors' firms these days), definitely check out his/her fees, as they can be substantially less than those charged by solicitors.