You might be able to do it, but even if you do, it might not work very well. To understand why, you'll need to understand a little about routing.
The purpose of a router is to move packets along the proper links, to get them closer to their destinations. But there are different types of routers that play different roles. There are the backbone routers, for instance, which form the core of the Internet; they take packets from many different places and send them to many different places, and for this purpose they maintain huge tables representing the structure of the entire (worldwide) backbone.
Then there are gateway routers like your DSL modem, which only need to know about the one network they sit on the boundary of. For any data going outside that network, your router will have a "default route", which effectively says, "If the address on this packet doesn't match any of the addresses for which I already have a more specific route [i.e. the addresses in this network], send it upstream," where "upstream" on a DSL modem usually means out to the DSL link and on toward your ISP. Unlike the aforementioned backbone routers, a DSL modem has a built-in notion of "upstream", and it's usually coded to always be on the DSL link.
Now, with some modems, it's possible to alter the routing table by hand so the default route points to an Ethernet port that you would set aside for the role. I know I could do that with mine, for instance, because it has a Linux-based firmware with a command-line shell option; so I could log in and use shell commands to change the routing table. But this change wouldn't survive a reboot, or a configuration change of any kind, for that matter; it's very much a "hack". Even if your modem allows this as well, I'd be surprised if it worked any differently in this respect.
A possible permanent solution would be to install a custom firmware that does what you want. The most obvious problem there is whether one even exists, and if it does, where you can find it. Then, of course, you have to trust it not to have some sort of malware. And there's the risk of bricking the modem if you make some sort of mistake, although it might have a recovery mode.
The easiest, and most likely best, solution is to get a standalone router. It may have desirable features you wouldn't have in the modem, and more importantly, it may have better security (assuming you choose correctly).