This to be possible with only 1 Arduino depends in your sensor. It is most likely not possibpe without another microcontroller (not nesessarily an Arduino) because of the following reason.
Most bluetooth modules on the market (linke the HC-05 vor HC-06) are bluetooth serial modules, meaning that they build a serial connection over bluetooth and provide the corresponding two pins (RX and TX). Yet most sensors don't implement an UART interface (the
Serial interface in Arduino), but only give an analog or digital output voltage. With much work you might be able to tweak the firmware of the bluetooth chip for reading and sending a digital value, but this would be more work, than the price of a second microcontroller would justify.
So I would try to add a small microcontroller (like an ATTiny85) to the sensors side. It can measure the sensors output voltage and then send it via
Serial to the bluetooth module. You might also want to use another Arduino here, though this might be an overkill. Maybe a really small version.
After some thinking I had the idea, that it might be possible to send an analog voltage from the sensor through a audio bluetooth module without using another Arduino. Though I'm not sure if this really would work and which module you need for that.
All in all I still would go for the extra microcontroller and serial interface bluetooth, because it is much easier for a beginner.
Above I mentioned using
Serial on an ATTiny85, which is not directly possible, since it does not have a full UART hardware (but only a USI). Though you can use the standard
SoftwareSerial library on it. If the rest of your code isn't to big, it should still fit in the small memory of the Tiny.