I doubt that anyone would consider Planet Ice to be 'built'. It's made of wood and tarpaulin and doesn't have a huge seating capacity. It's crap and was only supposed to be a temporary solution as the old ice rink was demolished to make way for the new St Davids.

There's a Cardiff Snowdome group on Facebook, but it all seems to have gone a bit quiet from them recently. However, this was posted on the group at the end of November:
'I can confirm that Cardiff Council is to shortly begin a procurement exercise to find a new development partner to build a potential new snow facility for the International Sports Village (ISV).
Obviously with the current financial climate being as it is, getting someone to deliver a snow facility is not going to be easy but this is a positive step which should be underway within the next two to three weeks.
The ISV has been a huge success story over the last few years with fantastic facilities such as the Cardiff International Pool and the Cardiff White Water centre opening and thriving.
The Council is committed to expanding the ISV and adding more attractions, of which the snow facility will hopefully be one.

Cllr Nigel Howells
Executive Member
Sport, Leisure & Culture
nhowells@cardiff.gov.uk'

There's also this article from Wales online.

I'd say, as a rough estimate, the dome will be built by 2014 at the latest if they can find someone to take the plans ahead. The South West is practically the only area of the UK that doesn't have easy access to a snowdome, so building one makes sense.

I would build it if I won the euromillions.