Griffiths believes the Cup final will be one of a number of major showpiece events to find a place at the new 72,000 all-seater Millennium Stadium.
Wembley is being knocked down and rebuilt next year and Griffiths will meet with Football Association commercial director, Philip Carling, today to push his bid for the 2001 FA Cup final. Griffiths said: "We have an infrastructure in Cardiff to stage such events. We are also looking to get the European Cup final in this stadium."
Griffiths has also targeted the Rugby League Cup final, major world boxing championships and rock concerts.
The WRU allowed the old National Stadium to host events like Wales' international football matches, soccer's Welsh Cup final, plus concerts including Michael Jackson and David Bowie.
Griffiths added: "We have the finest stadium in the world, there can be no doubt about that. We have a retractable roof covering it, which means that if it rains for the FA Cup final we can slide the roof across to keep the surface dry.
"We have already staged one big rugby international here and both Wales and South Africa were delighted with it. The FA Cup is not just for English clubs. We have three teams from Wales in the Nationwide in Cardiff, Swansea and Wrexham. We staged six major rugby internationals at Wembley. We used their facilities - now it's time to reciprocate."
However, the Millennium Stadium bid faces serious competition from other venues. Twickenham, Old Trafford, Villa Park, Anfield, Liverpool and Sunderland's Stadium of Light have been mentioned as possible rivals to Cardiff.
The Millennium Stadium will be completed for 1 October, when the ground will stage the opening ceremony of this year's World Cup, all of Wales' games in the competition, plus the final in November.
Griffiths added: "The rugby World Cup finals are just the beginning for the Millennium Stadium, which is at the heart of an ambitious city."
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