It depends if you have a power hungry processor or not. It's recommended that you have a 400 watt PSU for that card, but if you are running a processor that uses 65 watts or less, your power supply would probably be okay.
For a system using a single Radeon HD 4670 AGP graphics card AMD specifies a minimum of a 400 Watt or greater system power supply. The power supply should also have a maximum combined +12 Volt continuous current rating of 26 Amps or greater and have at least one available 4-pin Molex peripheral power connector.
AMD's power supply specs seem a little excessive since that card's worse case power draw should be less than 65 Watts when running FurMark.
What is the brand and model of your 350 Watt power supply unit?
What are the specifications of the rest of your system like CPU, motherboard, number of case cooling fans, etc? If your system is a consumer desktop PC (e.g. Acer, Compaq, Dell, Gateway, HP, ..., etc.) please specify its model name and number.