The BIOS is usually flashed in DOS. Sometimes BIOS vendors provide official tools to flash the BIOS from within Windows, or even Linux, but it's still safer to flash in a pure DOS environment. Fortunately it's quite easy to make a bootable DOS USB stick with the help of FreeDOS and Unetbootin, which is available for both Windows and Linux.
1 Linux
Please note that the instructions are meant for Fedora, or variants of it like the modified distribution pre-installed on the Acer Aspire One, but can be easily adapted to other distributions and package managment systems. So instead of yum use apt-get for Ubuntu or Debian. The package names are likely the same, and the packages may in fact already be installed.
Open a terminal. On the distribution pre-installed on the AA1 press Alt+F2 and check the checkbox to do so.
sudo yum install syslinux p7zip-plugins
wget -O unetbootin-bios http://downloads.sourceforge.net/unetbootin/unetbootin-linux-293
chmod 755 unetbootin-bios
sudo ./unetbootin-bios &
Select FreeDOS and the device node, just like in the screenshot, and click OK. The device node is usually /dev/sdb1 if only one USB device is connected.
1 Windows
Just download the file from here. Select FreeDOS and the drive letter of the USB drive, just like in the screenshot, and click OK.
2 Extract all files from the BIOS zip file onto the USB drive.
3 Reboot and press F12 to select a bootable device.
4 Choose option 5 when prompted by FreeDOS and ignore the error about a missing CD-ROM. Change to drive C: by typing C: and if that doesn't work try drive B: instead. If it then still doesn't work, or if FreeDOS doesn't even boot to a prompt, try a different USB drive as the AA1 can be a bit picky.
5 Run the batch file to flash the BIOS by simply entering its filename, which is 3310.BAT for the latest BIOS version. The process usually takes only a few seconds. The AC adapter must be connected or the flash utility will wait until it is.
6 After successfully flashing the BIOS the AA1 will reboot by itself. A few BIOS settings will be reset to default values and might need to be readjusted if they were altered before.