No current in homemade galvanic cell I built a galvanic cell using 0.01 M solutions of copper sulfate and zinc sulfate. I used copper and zinc electrodes and immersed them into the proper solutions (copper into copper sulfate, zinc into zinc sulfate). 

I used approximately 30 mL of each solution and put them into separate beakers. I connected the solutions using a salt bridge that was filled with 0.01 M potassium nitrate solution. I used cotton to cover the ends of the salt bridge (a plastic tube). 

I measured the voltage across the copper and zinc electrodes and I got around 1 V. However, there is no current being produced according to an ammeter. The galvanic cell cannot light up a small light bulb for instance. 

Furthermore, I took voltage measurements over a 50 minute period and the the voltage looked relatively constant throughout the period, though very slightly lower at the end.
Your small copper-zinc cell may not light up a bulb as often shown in textbooks. Check your ammeter settings, it must show some current. How are you connecting it in series? voltmeter is connected in parallel. Now an interesting question. Imagine you join 12 (e.g., Duracells) in series to generate about 12 Volts. Can one start a car with 12 Duracells? Why do you need a large car lead-acid battery? The answer lies in current. Same potential difference does not mean same amount of current supply. You may think 1 V is close to an ordinary cell, your single copper-zinc cell cannot supply enough current to even light up a bulb. Cell potential is a thermomdynamic concept. Current supply is a kinetic issue-how fast the electrons transfer to or from electrodes. Now you can think of the reason why there is no commercial battery made from zinc and copper. See Daniell cell for an interesting history. Check the size of electrodes in Daniell cells.