Can a dual-engine aircraft take off with only one engine?
It is a regulatory requirement (EASA CS 25) that commercial aircraft can safely fly should one of the two engines fail at any point of the flight, including takeoff at speeds beyond $V1$. (the maximum speed at which it can still stop before the end of the runway).


  Control Following Engine Failure  1 An acceptable means of showing
  compliance with CS 25.143(b)(1) is to demonstrate that it is possible
  to regain full control of the aeroplane without attaining a dangerous
  flight condition in the event of a sudden and complete failure of the
  critical engine in the following conditions:
  
  a. At each take-off flap setting at the lowest speed recommended for
  initial steady climb with all engines operating after take-off, with –
  i. All engines, prior to the critical engine becoming inoperative, at
  maximum take-off power or thrust;  ii. All propeller controls in the
  take-off position;  iii. The landing gear retracted;  iv. The
  aeroplane in trim in the prescribed initial conditions; and 
  
  b. With wing-flaps retracted at a speed of 1.23 VSR1 with – i. All
  engines, prior to the critical engine becoming inoperative, at maximum
  continuous power or thrust;  ii. All propeller controls in the
  en-route position;  iii. The landing gear retracted;  iv. The
  aeroplane in trim in the prescribed initial conditions.


Source



For a clip of this happening just as the aircraft is rotating, take a look at this video:


                
            

A simulator video of the procedure is also available: