Had two barclays cards from separate purchases, cancelled them both with no detriment to my credit score. But my situation is different than yours.
Normally, since you just got the card in March, its not providing a long credit history at this point. As long as it wasn't your first credit card, and you have enough of a credit limit on your other cards that overall balances don't exceed 25-30% of that limit, you "shouldn't" have a detriment either.
Except credit is a funny thing, since you only have one other card your credit utilization ratio is going to change greatly. (i.e., if your Chase card is a limit of $1000, Barclays is $2000, and you have a $500 balance on the Chase, canceling your barclays card takes you from 16% utilization to 50%, a large jump that will most definitely affect your score.). Even if you pay it off every month, the balance on your statement is what gets reported to the agencies.
boomhower says 1 Visa, 1 Mastercard, 1 AmEx or Discover, and I tend to agree with that. Its enough credit to show you are responsible, but not so much that a lender will think you could get into trouble really quickly by maxing them all out. In addition to the reasons boomhower provides, if you rotate their usage it also makes it more likely that all 3 credit reporting agencies are being reported to on a regular basis. Some banks only report to one or two, and you could get it in a situation where you have great reports at Experian and Equifax and nothing at TransUnion.
There are much better options than the Barclays card as a 'permanent' card, lower interest rates and some sort of rewards like cash back or airline miles for a starter.
3 cards also leaves room for a 'temporary' store card (like the Barclays) that you can apply for to get a special deal. For me, it makes it easier to track my toy purchases. i.e. I won't buy a new TV until the iPad is paid off. Mixing the toys in with everyday purchases makes it easy to lose track of how much you are really spending on non-necessities.
If you do cancel the Barclays card, I recommend that you do it after you have your second (or third) card in your hand. Make sure you have a 0 balance statement from Barclays, to make sure they don't ding you for $1.19 of interest somehow. If you have iTunes points or other rewards, make sure you claim them first otherwise they will be lost when the card is cancelled.
the myfico website has a lot of free info on how to raise and maintain your credit score. their forums have tons of real world people that may be in a similar situation
http://www.myfico.com/CreditEducation/ImproveYourScore.aspx