Summary form only given. We study the problem of energy-balanced data propagation in wireless sensor networks. The energy balance property guarantees that the average per sensor energy dissipation is the same for all sensors, during the entire execution of the protocol. This property is important since it prolongs the network's lifetime by avoiding early energy depletion of sensors. We propose a new algorithm that in each step decides whether to propagate data one-hop towards the final destination (the sink), or to send data directly to the sink. This randomized choice balances the (cheap) one-hop transmissions with the direct transmissions to the sink, which are more expensive but "bypass" the sensors lying close to the sink. Note that, in most protocols, these close to the sink sensors tend to be overused and die out early. By a detailed analysis we precisely estimate the probabilities for each propagation choice in order to guarantee energy balance. The needed estimation can easily be performed by current sensors using simple to obtain information. Under some assumptions, we also derive a closed form for these probabilities. The fact (shown by our analysis) that direct (expensive) transmissions to the sink are needed only rarely, shows that our protocol, besides energy-balanced, is also energy efficient.