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I had the ON card from August '06 until last month. I learned how to request CLI through this site and raised my CL by 4x what it was about a year ago. A long-time fan of ON, I always use my card at ON and reap the benefits (rewards). They offered me the Visa. Usually it takes over a year of card membership before they offer the Visa. It took over 5 years for me.As far as the account goes, your original account is closed and your new account is given a new age. RIght now, that may devastate your credit but in the long run, it's actually been helpful in the improvement of your creait.
From everything I've read about GE, you either have to be offered the VISA or you have to reapply for the card. Last January, when I wanted the Walmart Discover, (I have the store card) I was told by the rep to apply again and if I was eligible, they would send me the Discover. Well, I ended up with 2 Walmart store accounts. Ha-Ha... I ended up closing one and wasting the HP.
Start using your card a lot... I opened my BR store card in March 2011 ($300 CL). Used it a lot (including exchanges/returns) and they auto CLI me to $1000 on the 4th statement, and after the 7th statement I received an email inviting me to apply for the BR Visa (which they gave me for a $3000 CL). I attribute this 100% to using the card heavily (I order a lot of clothes online and do a lot of exchanges when they don't work out).
Also, they will close the store card when you open the Visa version. It's a soft pull only, but you will get the added account (hurting AAoA). You don't lose the history of the store card, though.