Several years later, and this question, and the answer, need more clarification.
Can you connect a vlan unaware router to a vlan aware switch?
YES. The router will belong to the untagged vlan at the port you connect it to.
Will it work out of the box?
Yes, the vlan aware switch will have a "default untagged vlan" assigned to all ports.
Will the router be able to talk to devices on both VLANs?
Not without a router. Out of the box, it will only talk to the devices that are on the default vlan. If you configure the port to be an access port to a different vlan.. then that would be the vlan that you have access to. You would need a vlan aware router on the vlan side of the equation to route traffic from the "internet's vlan" to the other vlan.
Can the PC's and the router be assigned to different vlans?
Yes, but with work. Their traffic will be assigned a default lan based on the access port (untagged vlan port) the router is connected to. However, There are ways to assign vlans based on your mac address. I know of the CISCO VMPS (vlan membership policy service), which VMPS compliant devices can use to ask the VMPS (server) for a VLAN based on the mac. Perhaps it has other ways such as based on the IP. Switches might have a setup were macs are explicitly setup via a table to be assigned to a vlan.
This setup can be configured to only accept one vlan (rejecting the others), accept multiple (if the mac is registered with a vlan, then there is no confusion as there is when there is no knowlege of the mac at all, accept unknown but have a default lan they'll be assigned to.. I think there's one more.
keep in mind that just because each machine is accepted to different vlans, they can only talk to each other when on the same vlan (the previous point).
"Thank you, so I understand there's no way around it and I must buy a
new router. I didn't buy a switch yet, I was sure the switch divides
the VLAN based on IP rather than ports"
The initial concept of a vlan allowed using multiple ports on a switch for different lans, creating virtual switches. The virtual switches switch virtual lans. But which particular port on a vlan can be decided by IP, or by Mac Address. As mentioned, a particular port on a switch can be assigned to a vlan by a lookup table or service.