Dripping weathercock spindle

[MisterBoy]

My octagonal summerhouse has a weathercock on the peak. Some sort of metal cap sits on the peak of the roof and there must be a gland or similar to let it spin.
Very similar to this:

In the very heavy rain over winter I've noticed it has started to drip inside, under the peak, but only when it's totally torrential.
What's my best option to stop this... Are we talking simply some grease or something else?

 

[Mw Roofline]

So usually the shaft running through the roof is fixed and the cock sits inside the shaft allowing it to spin so the leak should be around the shaft which I suppose you could use some decent mastic or perhaps lead sealant.