He made a statement following press suggestions that Australia were on standby to take over as hosts following concerns about preparations in the Caribbean.
Griffith said the World Cup would be discussed by the International Cricket Council (ICC) executive board meeting in Barbados over the next few days, adding: "WICB is fully aware of the significant challenges that it faces in hosting the 2007 World Cup.
"We are confident that as a nation we have the capacity to deliver on our obligations.
The Age, a Melbourne-based newspaper, has reported that Australia could take over the event if the ICC decided West Indies would not be ready to stage the showpiece.
The paper suggested concerns over the staging of the 2007 tournament in West Indies had reached a "significant level".
Griffiths said he did not know if comments attributed to Cricket Australia chairman Bob Merriman in the paper were accurate, adding: "We expect the fullest support and cooperation from all our fellow members of the ICC as we undertake our tasks."
Australia hosted the 1992 World Cup.
The proposed tournament in West Indies will be staged in 14 nations, with the possibility of matches also being played in Canada or the United States. There have been previous concerns expressed over travel, hotel accomodation and venues in the region.
ICC chief executive Malcolm Speed said earlier this year that it would be "inherently difficult" to stage the Cup in West Indies but that the sport's governing body was confident of it going ahead.