NASA have already designed an aircraft to do this, a modified 707 used for zero g training of space crew, popularly known as the 'Vomit Comet'.
By accelerating to Vne doing a 3g pull up to near vertical, the flight controller takes over and pushes the stick to give zero lift until its nearly in a Vne dive. It then repeats the process (hence the name).
With zero lift you experience 0g and hence follow a parabolic trajectory with the vertical speed falling at 10 m/s^2(about 20 knots/s) this is sustained for 20-30seconds.