Your old passport will have a hole drilled through it by the State Department, making it unusable for most purposes (i.e. most countries will take one look at it and say its invalid). If the visa itself is damaged, that could also make it invalid.
That said, it seems many countries will still honor visas inside old passports. The one time I dealt with this, I was told I should have the visa transferred asap to new passport. There was no fee for this but again, will vary. For folks with U.S. entry visas, it is a whole new application.
Best bet is to call the local Argentinian embassy or consulate. They have a bunch so see if one if physically nearby, and call that one (as I swear a lot of countries' policies seem to "vary" among consulates!). My guess is that since it isn't a visa per se, but showing payment of the reciprocity fee, you'd be fine carrying both passports. But worth verifying beforehand to avoid mishaps.
