iGoogle Gadgets cannot Sign In to Google Account

A while back, I noticed that two of my iGoogle gadgets had stopped working in Firefox on our Windows XP computer.  The Gmail gadget had stopped displaying any emails and the Calendar gadget had stopped displaying any appointments.  At the time I didn't particularly worry because I no longer relied on them for anything important.

Today I came across Gmail Tasks and decided to try out the iGoogle Tasks gadget.  It didn't work either.  It refused to accept that I was already signed into my Google account and kept prompting me to sign in again.

I had a look through my Firefox settings (Tools > Options > Privacy).  I already knew that iGoogle required cookies to be enabled and, sure enough, 'Accept cookies from sites' was ticked.  However, just below that was 'Accept third-party cookies' which was not ticked.  I ticked it and clicked 'OK'.

Now my Tasks, Gmail and Calendar gadgets all work as they should.