I've been jotting down some thoughts & designs on a handy napkin, and the main things I think we need to consider are the following:
1. Weight, I think the the ballon can lift around a maximum of 3kg usable payload. To get the maximum altitude after a motor burn we need the plane & control systems to be bare minimum of no more than 500 - 700gm weight. An iPhone for example is 140gm, so electronics & control mechanism are going to have to be 300gm at most. The Ardino board has sufficient avionics to do the job including an airspeed & barometric altimeter (more later).
2. Rocket Design, with 2.3kg of weight available for propellant / motors, the best option is to have a staged rocket motor assembly (ie Saturn 5 etc), this will give us the maximum burn time and potential altitude. By clustering 6 rocket motors in each stage and knozzle'ing them through a single chamber (shroud) we can overcome the vagiaries of motor ignition, so through the motors may not ignite symiltaneously they will be generally thrusting inline etc.
Refering to this link: http://www.thrustcurve.org/motorsearch.jsp (My maths is a bit rusty), but clustering 6 x 5n AeroTech E7 rockets together should thrust around 3kg of lift force for 5 seconds of burn time, at a weight of 50gm each motor, each cluster would have a weight of 300gm so very rough maths we have 700gm of plane & 7 x motor assembly stages + 200gm spare and a burn time of 30 seconds which should get us up to around 500 - 700mph and an extra 10,000 feet altitute so perhaps we can get to 90,000feet? http://webalt.markworld.com/multistage.html
3. Vertical launch; as long as the CoG is as far forward (nose) as posible and we have some drag to the rear of the rocket (ie fins). we should be able to launch using a fairly basic hook & eye mechanism. The avionics & barometric altimeter can be programmed to initialise the launch sequence before the baloon bursts, at which point the rocket assembly will be hanging in a reasonably vertical position (being that at 80,000 feet wind speed is going to be less chaotic, so any swing should be rasonably little) and as it shoots up into the inky blackness the hook drops from the baloon cable and we are gone!