Deleting the Ubuntu partitions and re-installing using the 64-bit Ubuntu installer should do the trick. It should also be possible to install the 64-bit Ubuntu directly over the old Ubuntu partitions. I don't have step-by-step instructions handy for either approach, since I don't recall every detail of the Ubuntu installer. The details of deleting partitions will also depend on the tool you use (GParted, parted, gdisk, etc.).

One caveat: If you're booting Windows in EFI mode (as is almost certainly the case if your computer came with Windows pre-installed), you probably want to do an EFI-mode installation of Ubuntu. You should check for an EFI-mode boot of the Ubuntu installation medium by looking for the directory /sys/firmware/efi. If it's present, you've booted in EFI mode. If it's absent, you've probably booted in BIOS/CSM/legacy mode. In the latter case, you should fiddle with your firmware settings and/or with your boot-time options to get the installer to boot in EFI mode. Note also that EFI-mode installations can still be trouble-prone because of buggy EFI implementations and less mature EFI support in Ubuntu. The number of potential problems is huge, which makes it impractical to describe all the possible problems; but fortunately, each such problem is rare, so there's a good chance you'll get through the process without incident. If you have problems, you'll just have to do a Web search or seek help on your specific problem. (IMHO, a forum is better for solving such problems than a site like this one, which is best for answering general questions that apply to many people.)

For additional detailed instructions, see UEFI in the Community Help.