This topic is very complex and you may here many different versions of the same idea because even historians do not agree. It is one of the few things in recent history that historians, all over the world, are so passionately divided about.
If I can say on the positive side, you may actually face less blatant discrimination, as a black person than say an immigrant from another Latin American country, simply because of the current mindset in Argentina.
With over 1 million Bolivians alone, not counting Paraguayans, Peruvians and quite a few others, illegal immigration is becoming a sensitive subject in the country because many people think the illegals are driving the country's social services bankrupt. On the other hand, without the immigrants (illegals or not), there would be literally no one to pick the crops in many places which would rot in the fields, or enough construction workers to keep wages somewhat in line in the current housing boom, or maids and nanny's for your middle class Argentine mother. So right now the focus of racist groups or individuals is the South American immigrant.
Last year a fire killed four bolivians working in almost slave-like conditions making clothes. It sent tens of thousands of immigrants to the streets in nationwide protests demanding legalization. With the large number of people from neighboring countries, the actual cases of real or extreme discrimination (physical threats or abuse), are not as high as they could be or once were. General discrimination though does yet exist, if you look mestizo with bronze skin, there are those that will call that person with racial slurs, of which there are many for different groups, religions, or nationalities.
Blacks are few in Argentina, yes, but not as rare as some claim. Specially these days in the largers cities there are more Jamaicans, Dominicans (specially women that were brought over with promises of work, but they were brought over for 'other' realities), and actual people from African nations, many refugees that come in on stowaway ships. There are also black Brazilians of course, and black Colombians (you will see several playing football/soccer in the Argentine League, or US blacks playing in the Argentine Basketball League). So it's not like you won't see another black person, but if you come from the USA or Brazil, it may seem that way!
The population of true black Argentines is not well known, the census does not count 'race' as a category in this country as it is against the constitution. One million is deffinately too much (the source of that figure being a pro-African group), but 10,000 might be too little. It also depends what is defined as 'black'. There actually could be quite a few Argentines that are 'octagoons' and the like, but the mix has been so diluted by wave after wave of non-African DNA that today they may not look black at all. Just like one time I met an American girl, a redhead with freckles that could have been from Ireland, but she was supposedly 1/8 black.
Will people gawk at you? Not everyone, but some probably will. And more out of simple curiosity, not that it diminishes the disconfort zone that may create.
It ultimately depends on what kind of person you are. If you can't tolerate any kind of strange looks or find motives behind every action, then you may have a rough time. But if you are the kind of person that just looks most things like that over, unless they become really egregious, then you really might have a good time and make some friends.
My internet was not shut down, the internet has shut me down