Decent HD cams for low light situations come at a price. Almost any are fantastic in daylight situations. My first mistake was buying what I thought to be a great prosumer camera, the HVR-HD1000u. Then I shot a wedding ceremony in a dim lit reception hall with no windows. After that I was about to shoot myself. Even "pro" cameras have their issues.
So options are, as stated above, pump out the light any way you can. This can get bulky, but you will want lights anyhow if you do a lot of shooting.
Only other option is to get a lower lux rated camera, but that will cost WAY more than lights. Plan on spending $3k+ on something that will handle low natural light "better". I know the HDR-fx1000 is a great option with manual gain control on it, almost another MUST if you film in low lights so you can limit the amount of maximum gain. This camera also has a minimum lux rating of 1.5, which is very good in the HD world.
For me with the wedding shooting, carrying lights around to all areas I need them on a moments notice is out of the question, so I HAD to upgrade my camera. My HDR-Z5U showed up yesterday. This is the same camera as the FX1000 I stated earlier, with mainly 2 more bells and whistles on it (XLR, and built in mount for a sony MRC1 CF recording unit) - but if you need to keep costs down, the FX1000 is the same camera with considerably lower price.
I am still learning this Z5U and will be for a long time to come, and will probably be posting questions of my own here shortly - but I must say the trip around my house last night with minimal sporadic lamp lighting turned out fantastic, so long as I kept manual reigns on my gain.
Of course if you really need to keep costs down, just stick with the lighting tips.