Mow the lawn you want to kill as short as possible. Mow on a dry day when rain is not forecast and do not irrigate your lawn after you mow it.
Flatten large cardboard boxes and remove all tape, staples and other objects. Alternatively, unfold double sheets of newspaper and create piles several layers thick.
Spread the flattened cardboard or thick layers of newspaper on top of the grass area you want to kill. Overlap pieces to completely cover the grass without leaving any areas of greenery or soil exposed.
Anchor the paper at the corners and seams using rocks or bricks.
Spread a 2 to 3 inch layer of topsoil on top of your paper covered area if you want to reseed immediately. Otherwise, leave the paper on your grassy area for several weeks until it is thoroughly dead before you work the soil for reseeding.