My iPhone keeps telling me that the email password is incorrect, when it's 100% correct.

I just got an iPhone 5c, upgraded from a 3GS-- on my 3GS, I could sign in via the Mail application to my Gmail with no problems; but now the 5c won't recognize the password. I've put the password into the mail app a dozen times, and I know for certain that it (as well as the email address itself) is correct. I've also tested it by logging into the web version of Gmail from Safari on my 5c, instead of the Mail application, and it works there as well as when I login from a computer. At this point I'm not sure if this is an Apple problem or a Gmail problem (I've sent this same query to Google for good measure), but it's rather stressful. This is my primary email account, and I'd like to get it working again.

EDIT: To clarify-- I have deleted and readded the email account in Settings, I have restarted the iPhone, I have done a backup to restore the previous settings... Nothing is changing. It keeps telling me that "The user name or password for 'imap.gmail.com' is incorrect.", and I know for fact that I'm typing it all in exactly as it should be.