Frodo's sword on Weathertop

[Lobelia]

If this topic has been discussed forgive me, but the search facility was slow tonight. I have been re-reading FOTR and have wondered what was going on with Frodo's sword, pre-Sting, I mean. During the fight against the Nazghul, he tries to stab the Witch-king and gets only his cloak, but I think his sword is broken at that point. Perhaps it was the Barrow-blade or perhaps not, but it broke, right? Yet only a few pages later, at the Ford of Bruinen, he is waving a sword defiantly at the Ring-Wraiths. What sword was this? Perhaps someone can tell me. Did he pack one he'd brought from home and then, sick or not, get it out again when the Barrow blade broke? Presumably he broke this one too, if so, because Bilbo gives him Sting to replace a broken sword Bilbo hadn't got around to having the Elves fix. Have I missed something between Weathertop and the ford?