Creating a library for arduino
Your problem stems from this line:

#define ADS1219


You define the macro ADS1219 as empty. From that point on all references to ADS1219 get replaced with nothing. For example:

class ADS1219{
public:
 ADS1219();
 void begin();
 void printHello();
};


becomes:

class {
public:
 ();
 void begin();
 void printHello();
};


And

void ADS1219::begin(){
 Serial.begin(9600);
}


becomes:

void ::begin(){
 Serial.begin(9600);
}


By using the same name in your header guard as your class name you screw everything else up.

I always use strict rules for header guard macros to ensure that they never conflict with anything else.  You should change yours to read something else.  I would use:

#ifndef _ADS1219_H
#define _ADS1219_H

// ... content

#endif




You are also lacking the parenthesis in the constructor implementation:

ADS1219::ADS1219{
 Serial.println("New object created!");
}


should read:

ADS1219::ADS1219(){
 Serial.println("New object created!");
}


And you don't give your ads variable a type and you don't want to use new:

ads = new ADS1219;


should read:

ADS1219 ads;