Would this computer work?

Mobo needs DDR3 RAM, you chose DDR2. Secondly, for your sanity, do not chose vista. XP is better all around, and Win7 is pretty awesome as well.
The rest of the parts will function together, yes, but there's issues:
The CPU / Mobo are yesterday's technology. MSI isn't a recommended brand for Mobo, ASUS and Gigabyte are better options. Rosewill PSU are not quality and should not be used. Look for a corsair unit, and you'll want more than 500W, look for 650+.

In my opinion, you should try and save up for a couple more weeks, get your budget around $700 and you can build a considerably higher quality, better performing, and more future proof system. I don't have links handy but I got bored and put together a solid system for about $750 on newegg.ca with AM3 dual core, sapphire 5770, gigabyte mobo with usb 3.0 / sata 6.0, WD 640GB sata 6.0 hdd, corsair 750tx, 4 GB corsair xms ram... $250 goes a very long way in this instance.

Athlon XII 255 3.1 GHz Dual Core CPU
Gigabyte GA-770TA-UD3 Bundled - 160.98 + 4.99 ship on CPU
Sapphire 5770 - 173.99 + 11.49 ship - 10.00 MIR
Antec 300 Illusion - 69.99
Sony DVD Burner - 27.99 + 6.99 ship
WD Caviar Black 640 GB 6.0 HDD - 79.99 + 7.24 ship
OCZ 4GB DDR3 1600 127.99 + 9.99 ship - 20.00 MIR
Corsair 750TX 108.99 - 10.00 MIR

This is the build I had come up with, totals:
749.92 + 40.70 ship - 40.00 MIR = 750.62

OS is going to run around $100 unless you can find a deal on a store clearing inventory of XP (again, do not get Vista). I found Win7 for $100, I'd prefer 32 bit personally, but for $25 less, I'd pick up the 64 bit unless you KNOW of something you're going to use that still has driver issues with the new OS and no workaround.

Pretty standard logitech keyboard and mouse - 18.99 + 7.99 ship
Win 7 64bit Home Premium - 99.99

This leaves you tight for a monitor after you add in the OS and keyboard. Newegg.ca doesn't seem to be quite as friendly as the US store, we get free shipping on lots of items and there are considerably more combo's that be found to save $20 here and there.
The only real place I can recall that you can cutback and save like $30 without sacrificing much would be dropping HDD to regular sata instead of 6.0. I got the mobo with this and usb 3.0 mainly for upgradeability a few years down the line when these are the new standards, and it's at a great price.
There are cheaper cases, although I wasn't particularly fond of any of them that were more than $10 cheaper than this one. You can get away with it, I just didn't feel they were of decent quality.

ASUS P5Q SE $110
http://www.newegg.ca/Product/Product.aspx?Item=N82E16813131347

OCZ Platinum 4GB (2 x 2GB) DDR3 1333 $125
http://www.newegg.ca/Product/Product.aspx?Item=N82E16820227478

Intel Pentium E6300 2.8 $93
http://www.newegg.ca/Product/Product.aspx?Item=N82E16819116091

COOLER MASTER Elite 360 RC $60
http://www.newegg.ca/Product/Product.aspx?Item=N82E16811119195

Use the other hardware suggested by hhnq04 and save $60.