"We have been advised that if my exH transfers his share of the house to me alone, we would not be liable to pay any Stamp Duty."
That is correct. It comes from Schedule 3, para 3 of the Finance Act 2003.
However, things can get complicated if you own more than one home.
If you own another home as well (eg a holiday home or investment property etc) then you will face the 5% stamp duty for people who own more than one home.
[Just as an aside, if your exH has already bought another house then he would not have had to pay the extra 5% as there is a special dispensation for divorced people who still own a share in a property but cannot sell the home due to a court order]
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I am confused by what you say here:
"The mortgage was paid off 15 years ago"
and
"...we will both contribute to the mortgage payments"
Do you mean that you will taking out a mortgage yourself in order to buyout your exH?
And you will then be doing a Deed of Trust stating that your boyfriend owns half of the house and will be paying whatever percentage of the mortgage? Will he be named on the mortgage?
This is where things get tricky.
If your boyfriend will be assuming any share of the mortgage, either by being named on the mortgage or having it mentioned in the Deed of Trust, then the share of the mortgage that is equivalent to his share of the house (so, if he owns half the house then he is taken to be responsible for half the mortgage) he takes on is subject to stamp duty (Finance Act 2003, Schedule 4, para 8(1)(b) ).
It will then depend on the size of the mortgage and whether or not your boyfriend is a first-time buyer.
I would really strongly recommend that you speak to somebody with specific knowledge of this area.