With a cold start - yes. But most GPS chips have memory backup for short-term almanac data (valid for up to 4 hours, works if the GPS doesn't move long distances in that time period) and internal clocks, so it can perform a warm start. This takes only a few micro Amps standby current, so it's not hard on the battery.

A SiRFstar with satellite prediction (SiRFInstantFixII) might be able to manage it.

5 seconds is too fast to acquire even a GSM fix to send the data.