root wrote:

The question is about securing a watertight connection between the
round side of the existing adapter and the round profile of the
existing outlet.

Right, so the downpipe is more or less where the two types of guttering
join - this makes it much clearer.

There is a cast iron running outlet and the square to round plastic adapter
sits on this.

Whatever you decide to use, it probably won't last very long, cast iron is
an arse to drill through and silicone doesn't adhere very well, my advice is
to splash out on some new square guttering and clips - I replaced some last
week.
The gutters were about £9 for a 5m length and the clips about £1.50 each,
end cap about £2 and running outlet about £4