Your v0 project might not be connected to your existing Vercel project. To deploy to your regular app URL:

1. Click the settings button (gear icon) in the top right corner
2. Go to "Settings"
3. Look for "Vercel project connection" - you can connect this v0 chat to your existing Vercel project
4. Once connected, publishing will deploy to your regular domain

If you already have a Vercel project with a custom domain, connecting it here will ensure future deploys go to that URL instead of a new auto-generated one.

I can see the Vercel project ID is set (prj_cS4XrrIKt73A9xNJZCCxp1blltyx), so it should be connected. The deployment issue is likely on the Vercel side, not something I can fix from here.

Try these steps:

1. In the Settings menu, click on the Vercel project connection and try disconnecting and reconnecting it
2. Check your Vercel dashboard directly at vercel.com/dashboard - look for the project and see if there are any failed deployments or errors
3. Try clicking Publish again after reconnecting

If it's still deploying to the wrong URL, the "Publish" button might be creating a new project instead of deploying to your connected one. In that case, you may need to:

1. Download the code (three dots menu > "Download ZIP" or use the shadcn CLI)
2. Push to your GitHub repo manually
3. Let Vercel auto-deploy from GitHub

Would you like me to help troubleshoot further, or would you prefer to go the GitHub route?