Can you eradicate a disease without finding a Cure first?
From the rules, page 4, when describing the Discover a Cure action:


  At any research station, discard 5 City cards of the same color from your
  hand to cure the disease of that color. Move the disease’s cure marker to its
  Cure Indicator.
  
  If no cubes of this color are on the board, this disease is now eradicated.
  Flip its cure marker from its “vial” side to its “blocked” side.


And, in the sidebar on that same page:


  Removing the last cube of a disease that is not cured has no effect.


Eradication is described as part of discovering a cure, so a cure is necessary for the disease to be eradicated. If you remove all of the cubes of the color, but don't have a cure, it is not eradicated and you place new disease cubes normally.

See also this thread on BGG, where Matt Leacock (one of the game's designers) weighed in:


  Eradicating a Disease
  
  If a cure for a given disease has been discovered and there are no disease cubes of that color on the board, immediately flip the Cure marker for the disease to the “Sunset” side. From now on, cards of this color have no effect when drawn on the Infector’s turn. 
  
  Take all of the cubes of the eradicated color and place them back in the box—they will not be used again for the rest of the game.