You'll only be "profitable" if you have a current system, and budget any hardware "investment" around a beefy graphics card.  The gpu is the component most directly related to mining- so building a whole new system when a current one that can accommodate a graphics card upgrade goes beyond splurging into, technically speaking, downright wasteful.
Do you have a desktop currently?  What are the specs?

check the bitcoin wiki for the bible of resources. here a some most relevant to your question:
check known tabulated values for hardware here:   https://en.bitcoin.it/wiki/Mining_hardware_comparison
plug-n-chug here:     https://en.bitcoin.it/wiki/Profitability_Calculator

since you're new and prospecting, do a lot of reading before purchasing!
I've done a bit of legwork for you by filling in a calculator with relevant values based on your question--> http://bitcoinx.com/profit/index.php?hashrate=400&cost=350&electricity=0.15&power=220&months=3.0000&decline=0.50
In that filled out calculator (see^), I've simplified and swung the numbers in your favor by quite a bit in terms of values and circumstance.
Look at the calculator I've filled in alongside while I explain why even with this handicapping of figures you'd be foolhearty to begin mining along the mindset of the approach you asked about:
I've assumed you spend that $350 on a new AMD 6970 placed in a current system- I'll reference this in the proceeding...
 

• The second box's value of 50BTC/block is a lofty, but unachievable (realistically)
• the 'hash rate' box I filled with a value attainable on a 6970 with a mild overclock
• in the next box I left the.15 cents/K=kWh but it depends on what you pay where you live (I have a rate of 13.5 cents/kWh)
• in filling in the pwr consumption box with 220W I've given you a huge break- this figure discounts the remaining non-gpu portion of your hypothetical computer. Also, 220W is a bit low considering the overclock to reach 400MH/s
• I left the timeframe at a quarter of a year
• $350 is ONLY the cost of a GPU NOT a new computer as you asked about

So even with a lot of wind to your back, unless you have deep pockets, its not really worth it considering 'break-even' is ~1.5 years.  I would even take that w/ a huge chunking grain of salt- a lot can happen in a year, even moreso that might be detrimental to an emerging 'digital currency'


I bought a amd 6950 for gaming, found out about bitcoins and mining for them, and although I have not actually converted my BTC earnings to cash... on paper I net roughly a dollar a day- maybe a bit more.  Back when I began, the BTC:USD exchange rate was ~300% loftier.  I would expect the valuation of bitcoins to slide as the internal mechanisms reduce mining profits and adoption either slows to a crawl, halts, reverses, or gets curb-stomped by the long arm of international/national governmental laws via crackdowns+taxation. 
I wouldn't approach bitcoin mining other than as a very lightly profitable use of a system you are utilizing as an ends in and of itself- my advice is not to invest in anything you don't stand a good chance of recouping in a short timeframe.  Maybe buy a open-box gfx card (saw a amd 5770 on newegg for ~50 dollars a week ago), let it start earning, and reinvest earnings to pay for improvements, rather than banking on the dumping of a large sum.  good luck