My 1997 Dodge Caravan has intermittent starting

My 1997 Dodge Caravan has intermittent starting problem. It will start but shut down in 2-3 seconds. When this happens, dash lights will not come on. (I can turn on accessory and know if van will start or not.) Yesterday had ignition switch replaced but its still happening. This happens whether it is hot, cold, wet or dry--no rhyme or reason. However, it does seem to happen more frequently when parked nose down on declining driveway as opposed to level parking on street.