Mignonette

We bought some mignonette seeds during the past winter and seem to be having trouble getting healthy plants. The seeds sprout, but become spindly and wilt and die.
My mother tried and a local florist tried in his greenhouse.
I'm sure we are not doing something conducive to good growth.

Mignonette is native in North Africa and presumably something in your conditions does not remind it sufficiently of home to be able to live! Hortus Third says that they like moderately rich soil in partly shaded locations and that after germination the night temperature should be kept at nine degrees Celsius or fourty-eight degrees Fahrenheit.
The seedlings should not be disturbed, because they transplant poorly.
I presume you have been using sterilized soil and containers for your seeding? There are many fungi just waiting to attack slowly growing seedlings! Look at the stem of the wilting seedlings and try to find a brown ring at the soil line. If one of the damping off fungi has damaged the seedflat the seedlings will slowly starve to death because that ring of dead tissue prevents the flow of nutrients from the food producing top to the raw material supplying roots.