Actually, the truth is a bit more complicated:

Mexico gained it's independence from Spain in 1822, the North of what used to be a Spanish colony never interested the new Mexican government.

America actually 'bought' the North half of Mexico from then president Santa Ana (yes, the same dude from El Alamo) when the US invaded Mexico the guy calling the shots for the US army was Santa Ana and not the US president, why ? (for 30 million reasons).

And for the Illegal problem it's actually a very complex one, on one side you have blatant corruption on the Mexican government since 1976, just take a look at the numbers: in early 1976 the exchange rate was 12.50 pesos to a dollar, right now it's 11,000 pesos to a dollar, that's an increase of 880 fold *(they did remove 3 zeroes from the peso)

Also the corruption problem extends to Central America and South America and add to that the need for cheap farm labor in the USA, like former Mexico president Fox said: 'Illegals come and do the work that not even blacks want to do in the US' sadly it's true, just go to the farm fields and check it out for yourself.

Another thing to consider is that when you give 'Papers' to an illegal he's not gonna go work at the farm field anymore, he's gonna want to have a city job or live of welfare or anything else he can get for free in the US.

Something else that's funny is the treatment the illegals get when caught by the Border Patrol, they process and deport them to Mexico, and guess who's waiting for them at the border ? of course the Mexican police, they grab them and steal their money (the one that the US border patrol didn't touch).