What I think would be a rough estimate (ignoring the physical/thermal requirements for diamond creation) TL;DR -- It's about 68W/m2 given the caveats below. Also:
Disclaimer: This is the best way I could think of to figure it out. It is a rough estimate only, and your level of desired precision will introduce more complicated calculations than the ones I've done. Not all of the calculations I performed are explicitly necessary, but were simply the best way for me to wrap my head around it at the time and may be incorrect!
Since plants are mostly glucose and water you could find the net glucose content of the plant (which will vary greatly -- but let's give a rough estimate of 90% as an average). So if you have a 10kg plant, ~1kg of the mass will be a mixture of cellulose, starch, and other stuff.
For the sake of ease, let's ignore the other stuff and focus on the cellulose and starch which we'll say makes up the 1kg. Both Cellulose and Starch are Glucose configurations, so what we have is 1kg of Glucose.
According to Wikipedia the net amount of energy a plant devotes to biomass construction/CO2 fixation is 3-6%, and the Gibbs Free Energy conversion of CO2 to Glucose is 114kcal. If I've done my math correctly that means a mole of CO2 to Glucose for a typical plant takes 2280kcal (or about what I burn in a day as a 6'4" man).
So, how many moles of Glucose is in our 1kg Glucose plant mass? The molecular mass of Glucose is 180.16g/mol. 1kg/180.16g/mol = 5.55 moles of Glucose. We don't need to know how many moles of CO2 we started with because we're not limited there; the atmosphere provides all the CO2 we could ask for. We just know that we have 5.55 moles of Glucose, and it took 12654kcal of energy to convert however many moles of CO2 into that many moles of Glucose.
That 12654kcal is also equivalent to 52944.34kJ of energy. Now, the solar constant is ~1.36kW/m2 ... or (1360J/s)/m2. So, how much area of your hypothetical plant is exposed to the Sun? A square-meter isn't an unreasonable amount for larger houseplants, so let's just work with that.
52944.34kJ / 1.36kW = 38929.66s = 648.82min = 10.8 hours' worth of Sun exposure to produce 1kg of plant mass. Some species of Bamboo can grow 91cm/day, but I couldn't find a source for density... but nearly 3-feet of bamboo could definitely weigh 1kg.
Sooooooo... Plants "take-in" 1.36kW/m2 when exposed to sunlight. How large the plant's area is will determine how much energy it takes in overall. How much energy a plant devotes to producing its mass is about 3-6%... or about 68 Joules every second (68 Watts using 5%) for a large plant with 1m2 exposed area.
Your 1ct diamond would take just over a minute and a half if 100% of the plant's mass-growing resources went into it.
Of course, that seems really fast. The important assumptions are that the plant is in direct sunlight, is not limited by other factors (Nitrogen is actually a major rate-of-growth limiter, as-is Phosphorous), halts all other growth-oriented metabolic activity to do so, and the metabolic processes for the conversion other carbon forms to Diamond was 100% efficient with the energy the plant demarcated for growth.
In reality about the only potentially "true" assumption from that list is that it's completely possible to have a plant with 1m2 of surface area exposed to sunlight. Everything else will be a fraction of what the working assumptions are.